GIFT  OF 
Dr.   Horace   Ivie 


^T^-r—ZJ^-^ 


ECLECTIC    EDUCATIONAL    SERIES. 


TREATISE 


GEOMETRY  AND  TRiadKOIETBt 


COLLEGES,  SCHOOLS,  AND   PRIVATE 
STUDENTS. 


WRITTEN   FOR  THE  MATHEMATICAL  COl'RSE  OF 

JOSEPH  RAY,  M.D., 


BY 

ELI  T.  TAPPAN,  M.A., 

P  R  0  F  E  S  S  0  K    OF    MATHEMATICS,    OHIO    UNIVERSITY. 


NEW-YORK     ..-.     CINCINNATI     .:•     CHICAGO 

AMERICAN    BOOK    COMPANY 

FROM    THK    PRESS  OF 
VAN    ANTWERP,    BRAGG,  A  CO. 


GIFT  OF  .       T^ 

Ray'S   Series, (i^^ 


EMBRACING 


A   Thorough  and  Progressive  Course  in  Arithmetic,  Algebra, 
and  the  Higher  Mathematics. 


Friiiiary  Aritliiuetic. 
lutiilectiial  AHtlwuftic. 
Itiidhnents  of  Ai'ithiuetie; 


Hig:li('r  Aritliiiiotie. 

Test  Examples  in  Arithmetic 

New  Elementary  Algebra. 


y.i'i'l'rii'.  AritJiMetJc.  I    New  Higher  Alg:el)ra. 


Plane  and  ^oli<l  €Jeoinotry.    By   Eli  T.  Tappan,  A.M..  Pres't 

Keayo)i  CoUege.     Vlino,  cloth,  27()  pp. 
€Je«iiietry    anil    Trigonometry.     By    Eli    T.   Tappax,   A.M. 

I'rcsU  Kcnjion    College.     8ro,  sheep,  420  pp. 
Analytic  Geometry.     By  Geo.  H.  Howison,  A.M.,  Prof,  in  Mass. 

Institute  of  Technology.    Treatise  on  Analytic  Geometry,  especially 

as   applied   to  the   Properties  of   Conies:    including   the   Modern 

Methods  of  Abridged  Notation. 
Eleinentis  of  Astronomy.    By  S.  H.  Peabody,  A.M.,  Prof,  of 

Physics  and  Civil  Engineering,  Amherst  CoUege.      Handsomely   and 

profusely  illustrated.     8ro,  sheep,  336  j)P- 


K  S  ITS. 

Ray'iS  Arithmetical  Key  ( To  Intellectual  and  Practical) ; 

l^ey  to  Ray'iii  Higher  Arithmetic : 

Key  to  Ray's  New  Elementary  and  Higher  Algebras. 


The  l*ubli.sh«'r.s  furnish  Descriptive  Circulars  of  the  above  Mathe- 
matical Text-Boohs,  with  I'rices  and  other  iufortnation  concerniiitj 
them. 


Entered  according  to  Act  of  Congress,  in   tlie  year  18G8,  by  Sargent,  Wilson  & 

HiNKi.E,  in  tlie  Clerk's  Office  of  tiie  District  Court  of  the  United 

States  for  the  Sontliern  District  of  Oliio. 

eOUCATION  DEPT 


PREFACE 


The  science  of  Elementary  Geometry,  after  remaining 
nearly  stationary  for  two  thousand  years,  has,  for  a  century 
past,  been  making  decided  progress.  This  is  owing,  mainly, 
to  two  causes:  discoveries  in  the  higher  mathematics  have 
thrown  new  light  upon  the  elements  of  the  science ;  and 
the  demands  of  schools,  in  all  enlightened  nations,  have 
called  out  many  works  by  able  mathematicians  and  skillful 
teachers. 

Professor  Hayward,  of  Harvard  University,  as  early  as 
1825,  defined  parallel  lines  as  lines  having  the  same  direc- 
tion. Euclid's  definitions  of  a  straight  line,  of  an  angle, 
and  of  a  plane,  were  based  on  the  idea  of  direction,  which 
is,  indeed,  the  essence  of  form.  This  thought,  employed  in 
all  these  leading  definitions,  adds  clearness  to  the  science 
and  simplicity  to  the  study.  In  the  present  work,  it  is 
sought  to  combine  these  ideas  with  the  best  methods  and 
latest  discoveries  in  the  science. 

By  careful  arrangement  of  topics,  the  theory  of  each  class 
of  figures  is  given  in  uninterrupted  connection.  No  attempt 
is  made  to  exclude  any  method  of  demonstration,  but  rather 
to  present  examples  of  all. 

The  books  most  freely  used  are,  "Cours  de  geometric 
elementaire,  par  A.  J.  H.  Vincent  et  M.  Bourdon ; "  "  Ge- 
ometric theorique  et  pratique,  etc.,  par  H.  Sonnet;"  "Die 

(iii) 


9S4225 


IV  PREFACE. 

reine  elementar-matliematik,  von  Dr.  Martin  Ohm ; "  and 
"Treatise  on  Geometry  and  its  application  to  the  Arts,  by 
Rev.  D.  Lardner." 

The  subject  is  divided  into  chapters,  and  the  articles  are 
numbered  continuously  through  the  entire  work.  The  con- 
venience of  this  arrangement  for  purposes  of  reference, 
has  caused  it  to  be  adopted  by  a  large  majority  of  writers 
upon  Geometry,  as  it  had  been  by  writers  on  other  scien- 
tific subjects. 

In  the  chapters  on  Trigonometry,  this  science  is  treated 
as  a  branch  of  Algebra  applied  to  Geometry,  and  the  trig- 
onometrical functions  are  defined  as  ratios.  This  method  has 
the  advantages  of  being  more  simple  and  more  brief,  yet 
more  comprehensive  than  the  ancient  geometrical  method. 

For  many  things  in  these  chapters,  credit  is  due  to  the 
works  of  Mr.  I.  Todhunter,  M.  A.,  St.  John's  College,  Cam- 
bridge. 

The  tables  of  logarithms  of  numbers  and  of  sines  and 
tangents  have  been  carefully  read  with  the  corrected  edi- 
tion of  Callet,  with  the  tables  of  Dr.  Schron,  and  with 
those  of  Babbage. 

ELI  T.  TAPPAN, 

Chic  University,  Jan.  1,  1868. 


CONTENTS. 


PAOB. 


PART   FIRST.— INTRODUCTOEY. 
CHAPTER   I. 

PRELIMINARY. 

Logical  Terms, 9 

General  Axioms, 11 

Ratio  and  Proportion, 12 

CHAPTER    II. 

THE  SUBJECT  STATED. 

Definitions, 17 

Postulates  of  Extent  and  of  Form, 19 

Classification  op  Lines, 22 

Axioms  of  Direction  and  of  Distance,      ....  23 

Classification  of  Surfaces, 24 

Division  of  the  Subject, 26 

PART   SECOND.— PLANE    GEOMETRY. 
CHAPTER   III. 

STRAIGHT  LINES. 

Problems, 28 

Broken  Lines, 31 

Angles, 32 


VI  CONTENTS. 

PAGE. 

Perpendicular  and  Oblique  Lines 38 

Parallel  Lines, 43 

CHAPTER    IV. 

CIRCUMFERENCES. 

General  Properties  op  Circumferences,      ...  52 

Arcs  and  Radii, 53 

Tangents, 58 

Secants, , 59 

Chords, 60 

Angles  at  the  Center, 64 

Intercepted  Arcs, ,    .     .     .  72 

Positions  of  Two  Circumferences, 78 

CHAPTER    V. 
TRIANGLES. 

General  Properties  of  Triangles, 85 

Equality  of  Triangles, 93 

Similar  Triangles, 101 

CHAPTER   VI. 

QUADRILATERALS. 

General  Properties  of  Quadrilaterals,  .  .  .  119 

Trapezoids, 122 

Parallelograms, 123 

Measure  of  Area, 128 

Equivalent  Surfaces, 135 

CHAPTER    VII. 
POLYGONS. 

General  Properties  of  Polygons, 143 

Similar  Polygons, 147 


CONTENTS.  VII 

Page. 

Regular  Polygons, 151 

isoperimetry, 159 

.  CHAPTER    VIII. 
CIRCLES. 

Limit  of  Inscribed  Polygons, 164 

Rectification  of  the  Circumference, 166 

Quadrature  of  the  Circle, 172 

PART  THIRD.— GEOMETRY   OF  SPACE. 

CHAPTER   IX. 
straight  lines  and  planes. 

Lines  and  Planes  in  Space, 177 

DiEDRAL  Angles, 185 

Parallel  Planes, 190 

Triedrals, 195 

polyedrals, 209 

CHAPTER    X. 
POLYEDRONS. 

Tetraedrons, 213 

Pyramids, 222 

Prisms, 226 

Measure  of  Volume, 232 

Similar  Polyedrons, 239 

Regular  Polyedrons, 241 

CHAPTER   XI. 
SOLIDS  OF  REVOLUTION. 

Cones, 247 

Cylinders, 249 


Viii  CONTENTS. 


PAGE. 


Spheres, 250 

Spherical  Areas, 261 

Spherical  Volumes, 270 

Mensuration, 276 

PART    FOURTH.— TRIGONOMETRY. 
CHAPTER    XII. 

PLANE  TRIGONOMETRY. 

Measure  of  Angles, 277 

Functions  of  Angles, 279 

Construction  and  Use  of  Tables, 296 

Right  angled  Triangles, 302 

Solution  of  Plane  Triangles, 304 

CHAPTER    XIII. 
SPHERICAL    TRIGONOMETRY. 

Spherical  Arcs  and  Angles, 314 

Right  angled  Spherical  Triangles,     .....     324 
Solution  of  Spherical  Triangles, 329 

CHAPTER    XIV. 

LOGARITHMS. 

Use  of  Common  Logarithms, 334 

TABLES. 

Logarithmic  and  Trigonometric  Tables,     .    .     .    345 


£LEMEJ>ftS' 


GEOMETRY. 


CHAPTER   I.— PRELIMINARY. 

Article  1.  Before  the  student  begins  the  study  of 
geometry,  he  should  know  certain  principles  and  defini- 
tions, which  are  of  frequent  use,  though  they  are  not 
peculiar  to  this  science.  They  are  very  briefly  pre- 
sented  in   this  chapter. 

LOGICAL    TERMS. 
3.  Every  statement  of  a  principle  is  called  a  Propc- 

SITIOX. 

Every  proposition  contains  the  subject  of  which  the 
assertion  is  made,  and  the  property  or  circumstance 
asserted. 

When  the  subject  has  some  condition  attached  to  it, 
the  proposition  is  said  to  be  conditional. 

The  subject,  with  its  condition,  if  it  have  any,  is  the 
Hypothesis  of  the  proposition,  and  the  thing  asserted 
is  the  CoNCLUSiox. 

Each  of  two  propositions  is  the  Converse  of  the  other, 
when  the  two  are  such  that  the  hypothesis  of  either  is 
the  conclusion  of  the  other. 

(9) 


10  ELKMENTS    OF    GEOMETRY. 

3.  A  proposition  is  either  fh'oretical,  that  is,  it  de> 
dares, that  a  certain  property  belongs  to  a  certain  thing; 
or  it  d,s*  jt?raf'^?va'(,:i?b(ai'Ji,is,  it  declares  that  something  can 
be ,  do.rie; 

'  •  P'rop'ositi'^ns  t^rc  citiicr  demonstrable^  that  is,  they  may 
b3  established  by  the  aid  of  reason ;  or  they  are  mdemon- 
sfrable,  that  is,  so  simple  and  evident  that  they  can  not 
be  made  more  so  by  any  course  of  reasoning. 

A  Theorem  is  a  demonstrable,  theoretical  proposition. 

A  Problem  is  a  demonstrable,  practical  proposition. 

An  Axiom  is  an  indemonstrable,  theoretical  propo- 
sition. 

A  Postulate  is  an  indemonstrable,  practical  propo- 
sition. 

A  proposition  Avhich  flows,  without  additional  reason- 
ing, from  previous  principles,  is  called  a  Corollary. 
This  term  is  also  frequently  applied  to  propositions, 
the  demonstration  of  which  is  very  brief  and  simple. 

4.  The  reasoning  by  which  a  proposition  is  proved 
is  called  the  Demonstration. 

The  explanation  how  a  thing  is  done  constitutes  the 
Solution  of  a  problem. 

A  Direct  Demonstration  proceeds  from  the  premises 
by  a  regular  deduction. 

An  Indirect  Demonstration  attains  its  object  by 
showing  that  any  other  hypothesis  or  supposition  than 
the  one  advanced  would  involve  a  contradiction,  or  lead 
to  an  impossible  conclusion.  Such  a  conclusion  may  be 
called  absurd,  and  hence  the  Latin  name  of  this  method 
of  reasoning — redudio  ad  absurdum. 

A  work  on  Geometry  consists  of  definitions,  proposi- 
tions, demonstrations,  and  solutions,  with  introductory 
or  explanatory  remarks.  Such  remarks  sometimes  have 
the  name  of  scholia. 


GEiNEKAL   AXIOMS.  U 

5,  Remark. — Tlie  student  should  learn  each  proposition,  so  as 
to  state  separately  the  hypothesis  and  the  conclusion,  also  the 
condition,  if  any.  He  should  also  learn,  at  each  demonstration, 
whether  it  is  direct  or  indirect;  and  if  indirect,  then  what  is  the 
false  hypothesis  and  what  is  the  absurd  conclusion.  It  is  a  good 
exercise  to  state  the  converse  of  a  proposition. 

In  this  work  the  propositions  are  first  enounced  in  general 
terms.  This  general  enunciation  is  usually  followed  by  a  particu- 
lar statement  of  the  principle,  as  a  fact,  referring  to  a  diagram. 
Then  follows  the  demonstration  or  solution.  In  the  latter  part 
of  the  work  these  steps  are  frequently  shortened. 

The  student  is  advised  to  conclude  every  demonstration  with  the 
general  proposition  which  he  lias  proved. 

The  student  meeting  a  reference,  should  be  certain  that  he  can 
state  and  apply  the  principle  referred  to. 


GENERAL   AXIOMS. 

6.   Quantities  which  are  each  equal  to  the  same  quan- 
tify, are  equal  to  each  other. 

T.  If  the   same   operation   be  performed   upon  equal 
quantities,  the  results  will  be  equal. 

For  example,  if  the' same  quantity  be  separately  added 
to  two  equal  quantities,  the  sums  will  be  equal. 

8.  If  the  same  operation  be  performed  upon  unequal 
quantities,  the  results  will  be  unequal. 

Thus,  if  the  same  quantity  be  subtracted  from  two 
unequal  quantities,  the  remainder  of  the  greater  will 
exceed  the  remainder  of  the  less. 

9.  The  tvhole  is  equal  to  the  sum  of  all  the  parts. 

1 0.  The  ivhole  is  greater  than  a  part. 

EXERCISE. 

11,  What  is  the  hypothesis  of  the  first  axiom  ?  Ans.  If  sev- 
eral quantities  are  each  equal  to  the  same  quantity. 


12  ELEMENTS   OF    GEOMETRY. 

What  is  the  subject  of  the  first  axiom  ?  Ayis.  Several  quan- 
tities. 

What  is  the  condition  of  the  first  axiom  ?  Ayis.  That  they  are 
each  equal  to  the  same  quantity. 

What  is  the  conclusion  of  the  first  axiom?  Ans.  Such  quan- 
tities are  equal  to  each  other. 

Give  an  example  of  this  axiom. 


RATIO    AND    PROPORTION 

12.  All  mathematical  investigations  are  conducted 
by  comparing  quantities,  for  we  can  form  no  conception 
of  any  quantity  except  by  comparison. 

13.  In  the  comparison  of  one  quantity  with  another, 
the  relation  may  be  noted  in  two  ways  :  either,  first, 
how  much  one  exceeds  the  other;  or,  second,  how  many 
times  one  contains  the  other. 

The  result  of  the  first  method  is  the  difference  be- 
tween the  two  quantities ;  the  result  of  the  second  is  the 
Ratio  of  one  to  the  other. 

Every  ratio,  as  it  expresses  "  how  many  times  "  one 
quantity  contains  another,  is  a  number.  That  a  ratio 
and  a  number  are  quantities  of  the  same  kind,  is  fur- 
ther shown  by  comparing  them;  for  Ave  can  find  their 
sum,  their  difference,  or  the  ratio  of  one  to  the  other. 

When  the  division  can  be  exactly  performed,  the  ratio 
is  a  whole  number;  but  it  may  be  a  fraction,  or  a  radical, 
or  some  other  number  incommensurable  Avith  unity. 

14.  The  symbols  of  the  quantities  from  whose  com- 
parison a  ratio  is  derived,  are  frequently  retained  in  its 
expression.     Thus, 

The  ratio  of  a  quantity  represented  by  a  to  another 

represented   by  5,  may  be  written  , . 

A  ratio  is  usually  written  a  :  h,  and  is  read,  a  is  to  b. 


RATIO    AND    PROPORTION.  13 

This  retaining  of  the  symbols  is  merely  for  conven- 
ience, and  to  show  the  derivation  of  the  ratio;  for  a 
ratio  may  be  expressed  by  a  single  figure,  or  by  any 
other  symbol,  as  2,  m^  ]/3,  or  ti.  But  since  every  ratio 
is  a  number,  therefore,  when  a  ratio  is  thus  expressed 
by  means  of  two  terms,  they  must  be  understood  to 
represent  two  numbers  having  the  same  relation  as  the 
given  quantities. 

The  second  term  is  the  standard  or  unit  with  which 
the  first  is  compared. 

So,  when  the  ratio  is  expressed  in  the  form  of  a  frac- 
tion, the  first  term,  or  Antecedent,  becomes  the  numera- 
tor, and  the  second,  or  Consequent,  the  denominator. 

15.  A  Proportion  is  the  equality  of  two  ratios,  and 
is  generally  written, 

a  :  h  '.  :  c  '.  d, 
and  is  read,        «  is  to  5  as  c  is  to  c?, 
but  it  is  sometimes  written, 

a  '.  h  =  c  :  d^ 

or  It  may  be,  h^d' 

all  of  which  express  the  same  thing:  that  a  contains  h 
exactly  as  often  as  c  contains  d. 

The  first  and  last  terms  are  the  Extremes,  and  the 
second  and  third  are  the  Means  of  a  proportion. 

The  fourth  term  is  called  the  Fourth  Proportional 
of  the  other  three. 

A  series  of  equal  ratios  is  written, 

a  :  h  :  :  c  :  d  :  :  e  :  f,  etc. 

When  a  series  of  quantities  is  such  that  the  ratio  of 
each  to  the  next  following  is  the  same,  they  are  written, 

a  :  h  :  c  :  d,  etc. 


14  ELEMENTS   OF   GEOMETRY. 

Here,  each  term,  except  the  first  and  last,  is  both  an- 
tecedent and  consequent.  When  such  a  series  consists 
of  three  terms,  the  second  is  the  Mean  Proportional 
of  the  other  two. 

16.  Proposition. —  The  product  of  the  extremes  of  any 
proportion  is  equal  to  the  product  of  the  meaiis. 

For  any  proportion,  as 

a  :  b  :  :  c  :  d, 
is  the  equation  of  two  fractions,  and  may  be  written, 

a c 

b^'d' 

Multiplying  these  equals  by  the  product  of  the  denom- 
inators, we  have  (7) 

aXd  =  bXc, 
or  the  product  of  the  extremes  equal  to  the  product  of 
the  means. 

17.  Corollary — The  square  of  a  mean  proportional 
is  equal  to  the  product  of  the  extremes.  A  mean  pro- 
portional of  two  quantities  is  the  square  root  of  their 
product. 

18.  Proposition. —  When  the  product  of  two  quanti- 
ties is  equal  to  the  product  of  two  others^  either  tivo  may  be 
the  extremes  and  the  other  two  the  means  of  a  proportion. 

Let  aXd=^bXe  represent  the  equal  products. 
If  we  divide  by  b  and  d,  we  have 

(1st.) 
(2.1.) 


a 
b~ 

'-%'  "'^ 

a  :  b  :  :  c  : 

d. 

If 

we 

divide  by 

c  and  d, 

we  have 

a_ 
e 

b 
=  d^    ^^' 

a  :  c  :  :  b 

:d. 

If 

we 

arrange  the  equal  products  thus: 

^-Xt'^ 

=  aXd, 

RATIO    AND    PROPORTION.  15 

and  then  divide  by  a  and  c,  we  have 

h:a::d'.c.  (3d.) 

By  similar  divisions,  the  student  may  produce  five 
other  arrangements  of  the  same  quantities  in  pro- 
portion. 

19.  Proposition The    order    of    the    terms    may    he 

changed  without  destroying   the  proportion,  so  long  as  the 
extremes  remain  extremes,  or  both  become  means. 

Let  a  :  b  :  :  c  :  d  represent  the  given  proportion. 

Then  (16),  we  have  aXd  =  bXc.  TherCxWe  (18),  a  and 
d  may  be  taken  as  either  the  extremes  or  the  means  of 
a  new  proportion. 

20.  When  Ave  say  the  first  term  is  to  the  third  as 
the  second  is  to  the  fourth,  the  proportion  is  taken  by 
alternation,  as  in  the  second  case.  Article  18. 

When  we  say  the  second  term  is  to  the  first  as  the 
fourth  is  to  the  third,  the  proportion  is  taken  inversely, 
as  in  the  third  case. 

21.  Proposition. — Ratios  which  are  equal  to  the  same 
ratio  are  equal  to  each  other. 

This  is  a  case  of  the  first  axiom  (6). 

22.  Proposition. — If  tivo  quantities  have  the  same 
multiplier,  the  multiples  will  have  the  same  ratio  m  the 
given  quantities. 

Let  a  and  b  represent  any  two  quantities,  and  m  any 
multiplier.     Then  the  identical  equation, 

7nXaXb  =  mXbXa, 
gives  the  proportion, 

mXa  :  mXb  ::  a:  b  (18). 

23.  Proposition. — In  a  series  of  cqiial  7'atios,  the  sum 
of  the  antecedents  is  to  the  suyn  of  the  consequents  as  any 
antecedent  is  to  its  consequent. 


16  ELEMENTS   OF   GEOMETRY. 

Let  a  :  b  ::  c  :  d  ::  e  :f  :  :  g  :  h,  etc.,  represent  the 
equal  ratios. 

Therefore  (16),  aXd  =  bXc 

aXf=bXe 

aXh  =  bXg 

To  these  add  aXb  =  bXa 

aX{b^-d-\-f^h)  =  bX{a-i-c-\-e+g), 
Therefore  (18), 

a^c-\-e-^g  :  b^d-\-f^h  ::  a  :  b. 
This  is  called  proportion  by  Composition. 

24,  Proposition. —  The  difference  bekveen  the  first  and 
second  terms  of  a  proportion  is  to  the  second,  as  the  dif- 
ference between  the  third  and  fourth  is  to  the  fourth. 

The  given  proportion, 

a  :  b  :  :  c  :  d, 

may  be  written,  h^^d' 

Subtract  the  identical  eqi^tion, 

b_d, 

b~d 
The  remaining  equation, 

a  —  b       c  —  d 

T'^'^dT' 

may  be  written,      a  —  b  :  b  :  :  c  —  d  :  d. 

This  is  called  proportion  by  Division. 

25.  Proposition — If  four  quantities  are  in  proportion, 
their  same  potvers  are  in  proportion,  also  their  same  roots. 


Thus,  if  we  have      a  :  b 
then,  a"^  :  6^ 

also,  i/a  :  |/5 


:  c  :  d, 

:  c":  d^; 
:  i/c  :  \/d. 


These   principles   are   corollaries  of  the  second  gen- 
eral axiom  (7),  since  a  proportion  is  an  equation. 


THE   SUBJECT   STATED.  }J 


CHAPTER     II. 
THE    SUBJECT    STATED. 

26-  We  know  that  every  material  object  occupies  a 
portion  of  space,  and  has  extent  and  form. 

For  example,  this  book  occupies  a  certain  space;  it 
has  a  definite  extent,  and  an  exact  form.  These  prop- 
erties may  be  considered  separate,  or  abstract  from  all 
others.  If  the  book  be  removed,  the  space  which  it  had 
occupied  remains,  and  has  these  properties,  extent  and 
form,  and  none  other. 

27.  Such  a  limited  portion  of  space  is  called  a  solid. 
Be  careful  to  distinguish  the  geometrical  solid,  which 

is  a  portion  of  space,  from  the  solid  body  which  occu- 
pies space. 

Solids  may  be  of  all  the  varieties  of  extent  and  form 
that  are  found  in  nature  or  art,  or  that  can  be  imagined. 

28.  The  limit  or  boundary  which  separates  a  solid 
from  the  surrounding  space  is  a  surface.  A  surface  is 
like  a  solid  in  having  only  these  two  properties,  extent 
and  form ;  but  a  surface  differs  from  a  solid  in  having 
no  thickness  or  depth,  so  that  a  solid  has  one  kind  of 
extent  which  a  surface  has  not. 

As  solids  and  surfaces  have  an  abstract  existence, 
without  material  bodies,  so  two  solids  may  occupy  the 
same  space,  entirely  or  partially.  For  example,  the 
position  which  has  been  occupied  by  a  book,  may  be  now 
occupied  by  a  block  of  wood.  The  solids  represented 
Geoni. — 2 


18  ELEMENTS   OF   GEOMETRT. 

by  the  book  and  block  may  occupy  at  once,  to  some  ex- 
tent, the  same  space.  Their  surfaces  may  meet  or  cut 
each  other. 

29.  The  limits  or  boundaries  of  a  surface  are  lines. 
The  intersection  of  two  surfaces,  being  the  limit  of  the 
pirts  into  which  each  divides  the  other,  is  a  line. 

A  line  has  these  two  properties  only,  extent  and  form  ; 
but  a  surface  has  one  kind  of  extent  which  a  line  has 
not:  a  line  differs  from  a  surface  in  the  same  way  that 
a  surface  does  from  a  solid.  A  line  has  neither  thick- 
ness nor  breadth. 

SO.  The  ends  or  limits  of  a  line  are  points.  The 
intersections  of  lines  are  also  points.  A  point  is  unlike 
either  lines,  surfaces,  or  solids,  in  this,  that  it  has  neither 
extent  nor  form. 

31.  As  one  line  may  be  met  by  any  number  of  oth- 
ers, and  a  surface  cut  by  any  number  of  others;  so  a 
line  may  have  any  number  of  points,  and  a  surface  any 
number  of  lines  and  points.  And  a  solid  may  have 
any  number  of  intersecting  surfaces,  with  their  lines 
and  points. 

DEFINITIONS. 

32.  These  considerations  have  led  to  the  followinjr 
definitions : 

A  Point  has  only  position,  without  extent. 
A  Line  has  length,  without  breadth  or  thickness. 
A   Surface  has   length  and   breadth,  without   thick- 
ness. 

A  Solid  has  length,  breadth,  and  thickness. 

33.  A  line  may  be  measured  only  in  one  way,  or,  it 
may  be  said  a  line  has  only  one  dimension.  A  surface 
has  two,  and  a  solid  has  three  dimensions.     We  can  not 


THE   POSTULATES.  19 

conceive  of  any  thing  of  more  than  three  dimensions. 
Therefore,  every  thing  which  has  extent  and  form  be- 
longs to  one  of  these  three  classes. 

The  extent  of  a  line  is  called  its  Length;  of  a  sur- 
face, its  Area;  and  of  a  solid,  its  Volume. 

34.  Whatever  has  only  extent  and  form  is  called  a 
Magnitude. 

Geometry  is  the  science  of  magnitude. 

Geometry  is  used  whenever  the  size,  shape,  or  posi- 
tion of  any  thing  is  investigated.  It  establishes  the 
principles  upon  which  all  measurements  are  made.  It 
is  the  basis  of  Surveying,  Navigation,  and  Astronomy. 

In  addition  to  these  uses  of  Geometry,  the  study  is 
cultivated  for  the  purpose  of  training  the  student's  pow- 
ers of  language,  in  the  use  of  precise  terms;  his  reason, 
in  the  various  analyses  and  demonstrations;  and  his 
inventive  faculty,  in  the  making  of  new  solutions  and 
demonstrations. 

THE    POSTULATES. 

35.  Magnitudes  may  have  any  extent.  We  may 
conceive  lines,  surfaces,  or  solids,  which  do  not  extend 
beyond  the  limits  of  the  smallest  spot  which  represents 
a  point ;  or,  Ave  may  conceive  them  of  such  extent  as  to 
reach  across  the  universe.  The  astronomer  knows  that 
his  lines  reach  to  the  stars,  and  his  planes  extend  be- 
yond the  sun.  These  ideas  are  expressed  in  the  fol- 
lowing 

Postulate  of  Extent. — A  magnitude  may  he  made  fo 
have  any  extent  tvhatever. 

36.  Magnitudes  may,  in  our  minds,  have  any  form, 
from  the  most  simple,  such  as  a  straight  line,  to  that 
of  the  most  complicated  piece  of  machinery.     Vy^o  may 


20  ELEMENTS    OF  GEOMETRY. 

conceive  of  surfaces  without  solids,  and  of  lines  witL 
surfiices. 

It  is  a  useful  exercise  to  imagine  lines  of  various 
forms,  extending  not  only  along  the  paper  or  blackboard, 
but  across  the  room.  In  the  same  way,  surfaces  and 
solids  may  be  conceived  of  all  possible  forms. 

The  form  of  a  magnitude  consists  in  the  relative  posi- 
tion of  the  parts,  that  is,  in  the  relative  directions  of  the 
points.  Every  change  of  form  consists  in  changing  the 
relative  directions  of  the  points  of  the  figure. 

Every  geometrical  conception,  however  simple  or  com- 
plex, is  composed  of  only  two  kinds  of  elementary 
thoughts — directions  and  distances.  The  directions  de- 
termine its  form,  and  the  distances  its  extent. 

Postulate  of  Form. —  The  points  of  a  magnitude  may  he 
made  to  have  from  each  other  any  directions  whatever ^  thus 
giving  the  magnitude  any  conceivable  form. 

These  two  are  all  the  postulates  of  geometry.  They 
rest  in  the  very  ideas  of  space,  form,  and  magnitude. 

37.  Magnitudes  which  have  the  same  form  while 
they  difter  in  extent,  are  called  Similar. 

Any  point,  line,  or  surface  in  a  figure,  and  the  simi- 
larly situated  point,  line,  or  surface  in  a  similar  figure, 
are  called  Homologous. 

Magnitudes  which  have  the  same  extent,  while  they 
differ  in  form,  are  called  Equivalent. 


MOTION  AND   SUPERPOSITION. 

38.  The  postulates  are  of  constant  use  in  geomet- 
rical reasoning. 

Since  the  parts  of  a  magnitude  may  have  any  posi- 
tion, they  may  change  position.     By  this   idea   of  mo- 


FIGURES.  21 

tion,  the   mutual   derivation    of   points,  lines,   surfaces, 
and  solids  may  be  explained. 

The  path  of  a  point  is  a  line,  the  path  of  a  line  may 
be  a  surface,  and  the  path  of  a  surface  may  be  a  solid. 
The  time  or  rate  of  motion  is  not  a  subject  of  geome- 
try, but  the  path  of  any  thing  is  itself  a  magnitude. 

39.  By  the  idea  of  motion,  one  magnitude  may  be 
mentally  applied  to  another,  and  their  form  and  extent 
compared. 

This  is  called  the  method  of  superposition,  and  is  the 
most  simple  and  useful  of  all  the  methods  of  demon- 
stration used  in  geometry.  The  student  will  meet  with 
many  examples. 

EQUALITY. 

40.  When  two  equal  magnitudes  are  compared,  it  is 
found  that  they  may  coincide;  that  is,  each  contains  the 
other.  Since  they  coincide,  every  part  of  one  will  have 
its  corresponding  equal  and  coinciding  part  in  the  other, 
and  the  parts  are  arranged  the  same  in  both. 

Conversely,  if  two  magnitudes  are  composed  of  parts 
respectively  equal  and  similarly  arranged,  one  may  be 
applied  to  the  other,  part  by  part,  till  the  wholes  coin- 
cide, showing  the  two  magnitudes  to  be  equal. 

Each  of  the  above  convertible  propositions  has  been 
stated  as  an  axiom,  but  they  appear  rather  to  constitute 
the  definition  of  equality. 


FIGURES. 

41.  Any  magnitude  or  combination  of  magnitudes 
which  can  be  accurately  described,  is  called  a  geomet- 
rical Figure. 


22  ELEMENTS    OF    GEOMETRY. 

Figures  are  represented  by  diagrams  or  drawings, 
and  such  representations  are,  in  common  language, 
called  figures.  A  small  spot  is  commonly  called  a 
point,  and  a  long  mark  a  line.  But  these  have  not  only 
extent  and  form,  but  also  color,  weight,  and  other  proper- 
ties; and,  therefore,  they  are  not  geometrical  points  and 
lines. 

It  is  the  more  important  to  remember  this  distinction, 
since  the  point  and  line  made  with  chalk  or  ink  are 
constantly  used  to  represent  to  the  eye  true  mathemat- 
ical points  and  lines. 

42.  The  figure  which  is  the  subject  of  a  proposition, 
together  with  all  its  parts,  is  said  to  be  Given.  The 
additions  to  the  figure  made  for  the  purpose  of  demon- 
stration or  solution,  constitute  the  Construction. 

43.  In  the  diagrams  in  this  work,  points  are  desig- 
nated   by    capital    letters.      Thus, 

the  points  A  and  B  are  at  the  ex- 
tremities of  the  line. 

Figures    are    usually  designated 
by  naming  some  of  their  points,  as 
the  line  AB,  and  the  figure  CDEF, 
or  simply  the  figure  DF. 


When  it  is  more  convenient  to  desig- 
nate a  figure  by  a  single  letter,  the 
small  letters  are  used.  Thus,  the  line 
a,  or  the  figure  h. 


LINES. 

44.  A  Straight  Line  is  one  which  has  the  same  di- 
rection throughout  its  whole  extent. 


A 

B 

c 

D 

\ 

\      - 

\ 

\ 

F 

a 

E 

THE    STRAIGHT    LINE.  23 

A  straight  line  may  be  regarded  as  the  path  of  a 
point  moving  in  one  direction,  turning  neither  up  nor 
down,  to  the  right  or  left. 

45.  A  Curved  Line  is  one  which  constantly  changes 
its  direction.  The  word  curve  is  used  for  a  curved 
line. 

46.  A  line  composed  of  straight 
lines,  is  called  Broken.  A  line 
may  be  composed  of  curves,  or  of 
both  curved  and  straight  parts. 

THE   STRAIGHT   LINE. 

47.  Problem. — A  straight  line  may  he  made  to  pass 
through  any  two  points. 

48.  Problem — There  may  he  a  straight  line  from  any 
pointy  in  any  direction,  and  of  any  exte7it. 

These  two  propositions  are  corollaries  of  the  post- 
ulates. 

49.  From  a  point,  straight  lines  may  extend  in  all 
directions.  But  we  can  not  conceive  that  two  separate 
straight  lines  can  have  the  same  direction  from  a  common 
point.     This  impossibility  is  expressed  by  the  following 

Axiom  of  Direction. — hi  one  direction  from  a  point, 
there  can  he  only  one  straight  line. 

50.  Corollary — From  one  point  to  another,  there  can 
be  only  one  straight  line 

51.  Theorem — If  a  straight  line  have  two  of  its  points 
commo7i  ivith  anolher  straight  line,  the  tivo  lines  must  coin- 
cide throughout  their  inutual  extent. 

For,  if  they  could  separate,  there  would  be  from  the 
point  of  separation  two  straight  lines  having  the  same 
direction,  which  is  impossible  (49). 


24  ELEMENTS    OF    GEOMETRY. 

52.  Corollary. — Two  fixed  points,  or  one  point  and 
a  certain  direction,  determine  the  position  of  a  straight 
line. 

53.  If  a  straight  line  were  turned  upon  tw^o  of  its 
points  as  fixed  pivots,  no  part  of  the  line  would  change 
place.  So  any  figure  may  revolve  about  a  straight  line, 
while  the  position  of  the  line  remains  unchanged. 

This  property  is  peculiar  to  the  straight  line.  If  the 
curve  BC  were  to  revolve  upon 
the  two  points  B  and  C  as  piv- 
ots, then  the  straight  line  con- 
necting these  points  would  remain  at  rest,  and  the  curve 
would  revolve  about  it. 

A  straight  line  about  which  any  thing  revolves,  is 
called  its  Axis. 

54.  Axiom  of  Distance — The  straight  line  is  the 
shortest  which  can  join  tivo  points. 

Therefore,  the  distance  from  one  point  to  another  is 
reckoned  along  a  straight  line. 

55.  There  have  now  been  given  two  postulates  and 
two  axioms.  The  science  of  geometry  rests  upon  these 
four  simple  truths. 

The  possibility  of  every  figure  defined,  and  the  truth 
of  every  problem,  depend  upon  the  postulates. 

Upon  the  postulates,  with  the  axioms,  is  built  the 
demonstration  of  every  principle. 

SURFACES. 

56.  Surfaces,  like  lines,  are  classified  according  to 
their  uniformity  or  change  of  direction. 

A  Plane  is  a  surface  which  never  varies  in  direction. 
A  Curved  Surface  is  one  in  which  there  is  a  change 
of  direction  at  every  point. 


THE   PLANE.  25 

THE    PLANE. 

5T.  The  plane  surface  and  the  straight  line  have  the 
same  essential  character,  sameness  of  direction.  The 
plane  is  straight  in  every  direction  that  it  has. 

A  straight  line  and  a  plane,  unless  the  extent  be 
specified,  are  always  understood  to  be  of  indefinite 
extent. 

58.  Theorem — A  straight  line  which  has  tivo  points  in 
a  plane,  lies  wholly  in  it,  so  far  as  they  both  extend. 

For  if  the  line  and  surface  could  separate,  one  or  the 
other  would  change  direction,  which  by  their  definitions 
is  impossible. 

59.  Theorem Two  planes  having  three  points  com- 
mon, and  not  in  the  same  straight  line,  coincide  so  far  as 
they  both  extend. 

Let    A,    B,    and    C    be    three 
points    which     are     not    in    one  F,,-x 

straight  line,  and  let  these  points     ^  ,^.-.::r.'r.*....\ \c 

be  common  to  two  planes,  which  ^\ 

may  be  designated  by  the  letters  \ 

m  and   p.      Let  a   straiorht   line  \^ 

pass   through   the   points  A  and 

B,  a  second  through  B  and  C,  and  a  third  through  A 

and  C. 

Each  of  these  lines  (58)  lies  wholly  in  each  of  the 
planes  m  and  p.  Now  it  is  to  be  proved  that  any  point 
D,  in  the  plane  m,  must  also  be  in  the  plane  p. 

Let  a  line  extend  from  D  to  some  point  of  the  line 
AC,  as  E.  The  points  D  and  E  being  in  the  plane  m, 
the  whole  line  DE  must  be  in  that  plane ;  and,  therefore, 
if  produced  across  the  inclosed  surface  ABC,  it  will  meet 
one  of  the  other  lines  AB,  BC,  which  also  lie  in  that 
plane,  say,  at  the  point  F.  But  the  points  F  and  E 
Geom. — 3 


26  ELEMENTS    OF    GEOMETRY. 

are    both    in    the   plane  p.     Therefore,  the  whole   line 
FD,  including  the  point  D,  is  in  the  plane  p. 

In  the  same  manner,  it  may  be  shown  that  any 
point  which  is  in  one  plane,  is  also  in  the  other,  and 
therefore  the  two  planes  coincide. 

60.  Corollary. — Three  points  not  in  a  straight  line, 
or  a  straight  line  and  a  point  out  of  it,  fix  the  position 
of  a  plane. 

61.  Corollary — That  part  of  a  plane  on  one  side 
of  any  straight  line  in  it,  may  revolve  about  the 
line  till  it  meets  the  other  part,  when  the  two  will 
coincide  (53). 

EXERCISES. 

62.  When  a  mechanic  wishes  to  know  whether  a  line  is 
straight,  he  may  apply  another  line  to  it,  and  observe  if  they 
coincide. 

In  order  to  try  if  a  surface  is  plane,  he  applies  a  straight  rule 
to  it  in  many  directions,  observing  whether  the  two  touch 
throughout. 

The  mason,  in  order  to  obtain  a  plain  surface  to  his  marble, 
applies  another  surface  to  it,  and  the  two  are  ground  together 
until  all  unevenness  is  smoothed  away,  and  the  two  touch 
throughout. 

"What  geometrical  principle  is  used  in 
each  of  these  operations  ? 

In  a  diagram  two  letters  suffice  to  mark 
a  straight  line.     Why? 

But  it  may  require  three  letters  to  designate  a  curve.     Why  ? 


DIVISION    OF    SUBJECT. 

03.  By  combinations  of  lines  upon  a  plane.  Plane 
Figures  are  formed,  which  may  or  may  not  inclose  an 
area. 

By  combinations   of  lines   and    surfaces,  figures  are 


DIVISION    OF    SUBJECT.  27 

formed  in  space,  which  may  or  may  not  inclose  a  vol- 
ume. 

In  an  elementary  work,  only  a  few  of  the  infinite  va- 
riety of  geometrical  figures  that  exist,  are  mentioned, 
and  only  the  leading  principles  concerning  those  few. 

Elementary  Geometry  is  divided  into  Plane  Geome- 
try, which  treats  of  plane  figures,  and  Geometry  in 
Space,  which  treats  of  figures  whose  points  are  not  all 
in  one  plane. 

In  Plane  Geometry,  we  will  first  consider  lines  with- 
out reference  to  area,  and  afterward  inclosed  figures. 

In  Geometry  in  Space,  we  will  first  consider  lines 
and  surfaces  which  do  not  inclose  a  space ;  and  after- 
v^^ard,  the  properties  of  certain  solids. 


28  ELEMENTS    OF    GEOMETRY. 


PLANE    GEOMETRY. 


CHAPTER    III. 

STRAIGHT    LINES. 

f»4.  Problem — Straight  lines  may  he  added  together^ 
and  one  straight  line  may  he  suhtracted  from  another. 

For  a  straight  line  may  be  produced  to  any  extent. 
Therefore,  the  length  of  a  straight  line  may  be  increased 
by  the  length  of  another  line,  or  two  lines  may  be 
added  together,  or  we  may  find  the  sum  of  several 
lines  (35j. 

Again,  any  straight  line  may  be  applied  to  another, 
and  the  two  will  coincide  to  their  mutual  extent.  One 
line  may  be  subtracted  from  another,  by  applying  the 
less  to  the  greater  and  noting  the  difference. 

65.  Problem — A  straight  line  may  he  multiplied  hy 
any  number. 

For  several  equal  lines  may  be  added  together. 

GO.  Problem — A  straight  line  may  he  divided  by 
another. 

By  repeating  the  process  of  subtraction. 

67.   Problem. — A   straight  line  may  he  decreased  in 
any  ratio.,  or  it  may  he  divided  into  several  equal  parts. 
This  is  a  corollary  of  the  postulate  of  extent  (35). 


PROBLEMS    IN    DRAWING.  29 

PROBLEMS    IN    DRAWING. 

68.  Exercises  in  linear  drawing  afford  the  best  applications  of 
the  principles  of  geometry.  Certain  lines  or  combinations  of  lines 
being  given,  it  is  required  to  construct  other  lines  which  shall 
have  certain  geometrical  relations  to  the  former. 

Except  the  paper  and  pencil,  or  blackboard  and  crayon,  the 
only  instruments  used  are  the  ruler  and  compasses;  and  all  the 
required  lines  must  be  drawn  by  the  aid  of  these  only.  The 
reason  for  this  rule  will  be  shown  in  the  following  chapter. 

The  ruler  must  have  one  edge  straiglit.  The  compasses  have 
two  legs  with  pointed  ends,  which  meet  when  the  instrument  is 
shut.  For  blackboard  work,  a  stretched  cord  may  be  substituted 
for  the  compasses. 

69,  With  the  rxiler^  a  straight  line  may  be  drawn  on  any  plane 
surface,  by  placing  the  ruler  on  the  surface  and  drawing  the  pen- 
cil along  tiie  straight  edge. 

A  straight  line  may  be  drawn  through  any  two  points,  after 
placing  the  straight  edge  in  contact  with  the  points. 

A  terminated  straight  line  may  be  produced  after  applying  the 
straight  edge  to  a  part  of  it,  in  order  to  fix  the  direction. 

f  O,  With  the  compasses^  the  length  of  a  given  line  may  be 
taken  by  opening  the  legs  till  the  fine  points  are  one  on  each  end 
of  the  line.  Then  this  length  may  be  measured  on  the  greater 
line  as  often  as  it  will  contain  the  less.  A  line  may  thus  be 
produced  any  required  length. 

71.  The  student  must  distinguish  between  the  problems  of 
geometry  and  problems  in  drawing.  The  former  state  what  can 
be  done  with  pure  geometrical  magnitudes,  and  their  truth  de- 
pends upon  showing  that  they  are  not  incompatible  with  the 
nature  of  the  given  figure;  for  a  geometrical  figure  can  have  any 
conceivable  form  or  extent. 

The  problems  in  drawing  corresponding  to  those  above  given, 
except  the  last,  "  to  divide  a  given  straight  line  into  proportional 
or  equal  parts,"  are  solved  by  the  methods  just  described. 

72.  The  complete  discussion  of  a  problem  in  drawing  includes, 
besides  the  demonstration  and  solution,  the  showing  whether  the 
problem  lias  only  one  solution  or  several,  and  the  conditions  of 
each. 


30  ELEMENTS    OF    GEOMETRY. 

STKAIGHT   LINES    SIMILAR. 

TS.  Theorem. — Any  two  straight  lines  are  similar  fig- 
ures. 

For  each  has  one  invariable  direction.  Hence,  two 
straight  lines  have  the  same  form,  and  can  differ  from 
each  other  only  in  their  extent  (37). 

74.  Any  straight  line  may  be  diminished  in  any 
ratio  (67),  and  may  therefore  be  divided  in  any  ratio. 

The  points  in  two  lines  which  divide  them  in  the  same 
ratio  are  homologous  points,  by  the  definition  (37). 
Thus,   if  the   lines  AB 

and  ED  are  divided  at  ^ 2 ? 

the  points  C  and  F,  so      jj  F  j) 

that  AC  :  CB  ::  EF  :  FD, 

then  C  and  F  are  homologous,  or  similarly  situated 
points  in  these  lines ;  AC  and  EF  are  homologous  parts, 
and  CB  and  FD  are  homologous  parts. 

75.  Corollary — Two  homologous  parts  of  two  straight 
lines  have  the  same  ratio  as  the  two  whole  lines. 

For,      AC+CB  :  EF+FD  :  :  AC  :  EF  (23). 
That  is,  AB  :  ED  :  :  AC  :  EF. 

Also,  AB  :  ED  :  :  CB  :  FD. 

76.  Problem  in  Drawing. — To  find  the  ratio  of  tivo 
given  straight  lines. 

Take,    for    example,  the 
lines  b  and  c.  

If  these  two  lines  have 
a  common  multiple,  that  is,  a  line  which  contains  each  of  them 
an  exact  number  of  times,  let  x  be  the  number  of  times  that  b  is 
contained  in  the  least  common  multiple  of  the  two  lines,  and  y 
the  number  of  times  it  contains  c.  Tlien  x  times  b  is  equal  to  ?/ 
Limes  c. 


BROKEN    LINES.  3  J 

Therefore,  from  a  point  A,  draw  an  indefinite  straight  line  AE. 


Apply  each  of  the  given  lines  to  it  a  number  of  times  in  sue- 
ression.  The  ends  of  the  two  lines  will  coincide  after  x  applica- 
tions of  b,  and  y  applications  of  c. 

If  the  ends  coincide  for  the  first  time  at  E,  then  AE  is  the  least 
common  multiple  of  the  two  lines. 

The  values  of  x  and  y  may  be  found  by  counting,  and  these 
express  the  ratio  of  the  two  lines.  For  since  y  times  c  is  equal  to 
X  times  b,  it  follows  that  b  :  c  :  :  y  :  x,  which  in  this  case  is  as 
3  to  5. 

It  may  happen  that  the  two  lines  have  no  common  multiple. 
In  that  case  the  ends  will  never  exactly  coincide  after  any  number 
of  applications  to  the  indefinite  line;  and  the  ratio  can  not  be  ex- 
actly expressed  by  the  common  numerals. 

By  this  method,  however,  the  ratio  may  be  found  within  any 
desired  degree  of  approximation. 

Iflf,  But  this  means  is  liable  to  all  the  sources  of  error  that 
arise  from  frequent  measurements.  In  practice,  it  is  usual  to 
measure  each  line  as  nearly  as  may  be  with  a  comparatively  small 
standard.     The  numbers  thus  found  express  the  ratio  nearly. 

Whenever  two  lines  have  any  geometrical  dependence  upon 
each  other,  the  ratio  may  be  found  by  calculation  with  an  accu- 
racy which  no  measurement  by  the  hand  can  reach. 


BROKEN    LINES. 

T8.  A  curve  or  a  broken  line  is  said  to  be  Concave 
on  the  side  toward  the  straight  line  which  joins  two  of 
its  points,  and  Convex  to  the  other  side. 

TO.  Theorem. — A  broken  line  which  is  convea-  toward 
another  line  that  unites  its  extreme  points,  is  shorter  than 
that  line. 

The  line  ABCD  is  shorter  than  the  line  AEGD,  to- 
ward which  it  is  convex. 


32  ELEMENTS   OF   GEOMETRY. 

Produce  AB  and  BC  till  they  meet  the  outer  line 
in  F  and   H. 

Since  CD  is  shorter  than  CHD, 
it  follows  (8)  that  the  line  ABCD 
is  shorter  than  ABHD.  For  a  simi- 
lar reason,  ABHD  isi  shorter  than 
AFGD,  and  AFGD  is  shorter  than  AEGD.  There- 
fore, ABCD  is  shorter  than  AEGD. 

The  demonstration  would  be  the  same  if*  the  outer 
line  were  curved,  or  if  it  were  partly  convex  to  the 
inner  line. 

EXERCISE. 

80.  Vary  the  above  demonstration  by  producing  tlie  lines  DC 
and  CB  to  the  left,  instead  of  AB  and  BC  to  the  right,  as  in  the 
text;   also, 

By  substituting  a  curve  for  the  outer  line;  also, 

By  letting  the  inner  line  consist  of  two  or  of  four  straight  lines. 

81,  A  fine  thread  being  tightly  stretched,  and  thus  forced  to 
assume  that  position  which  is  the  shortest  path  between  its  ends, 
is  a  good  representation  of  a  straight  line.  Hence,  a  stretched 
cord  is  used  for  marking  straight  lines. 

The  word  straight  is  derived  from  ^'- stretch^''  of  which  it  is  an 
obsolete  participle. 

ANGLES. 

82,  An  Angle  is  the  difference  in  direction  of  two 
lines  which  have  a  common  point. 

83.  Theorem. —  The  two  lines  which  form  an  angle 
lie  in  one  planer  and  determine  its  position. 

For  the  plane  may  pass  through  the  common  point 
and  another  point  in  each  line,  making  three  in  all. 
These  three  points  determine  the  position  of  the  plane 
(60).       . 


•>^;iiai^> 


ANGLES.  33 

DEFINITIONS. 

H4.  Let  the  line  AB  be  fixed,  and  the  line  AC  revolve 
m    a    plane    about    the    point    A; 
thus  taking  every  direction    from 
A  in  the   plane  of  its  revolution. 
The  angle   or  difference  in   direc- 
tion   of    the    two    lines    will    in-         -^  ^ 
crease  from  zero,  when  AC  coincides  with  AB,  till  AC 
takes  the  direction  exactly 
opposite  that  of  AB.                        ..  \  \  \  j  /  /  /  .. 

If  the  motion  be  contin-  "C VM\ I //vvv-'' x'' 

ued,  AC  will,  after  a  com- 
plete revolution,  again  co-       C 
incide  with  AB. 

The  lines  which  form  an  angle  are  called  the  Sides, 
and  the  common  point  is  called  the  Vertex. 

The  definition  shows  that  the  angle  depends  upon  the 
directions  only,  and  not  upon  the  length  of  the  sides. 

85.  Three  letters  may  be  used  to  mark  an  angle, 
the  one  at  the  vertex  being  in  the 

middle,  as  the  angle  BAC.  When 
there  can  be  no  doubt  what  angle 
is  intended,  one  letter  may  answer, 
as  the  angle  C.  ^ 

It  is  frequently  convenient  to  mark  angles 
with  letters  placed  between  the  sides,  as  the 
angles  a  and  b. 

Two  angles  are  Adjacent  when  they  have  the  same 
vertex  and  one  common  side  between  them.  Thus,  in 
the  last  figure,  the  angles  a  and  b  are  adjacent;  and,  in 
the  previous  figure,  the  angles  BAC  and  CAD. 

86.  A  straight  line  may  be  .regarded   as  generated 


S4  ELEMENTS    OF    GEOMETRY. 

by  a  point  from  either  end  of  it,  and  therefore  every 
straight  line  has  two  directions,  which  are  the  opposite 
of  each  other.  We  speak  of  the  direction  from  A  to  B 
as  the  direction  AB,  and  of  the  direction  from  B  to  A 
as  the  direction  BA. 

One  line  meeting  another  at  some  other  point  than 
t:ie  extremity,  makes  two  angles  B 

with  it.  Thus  the  angle  BDF  is 
the  difference  in  the  directions 
DB  and  DF ;  and  the  angle  BDC       C  D  F 

is  the  difference  in  the  directions  DB  and  DC. 

When  two  lines  pass  through  or  cut  each  other,  four 
angles  are  formed,  each  direction  of  one  line  making  a 
difference  with  each  direction  of  the  other. 

The  opposite  angles  formed  by  two  lines  cutting  each 
other  are  called  Vertical  angles. 

A  line  which  cuts  another,  or  which  cuts  a  figure,  is 
called  a  Secant. 

PROBLEMS    ON"    ANGLES. 

87.  Angles  may  be  compared  by  placing  one  upon 
the  other,  when,  if  they  coincide,  they  are  equal. 

Problem. — One  angle  may  he  added  to  another. 

Let  the  angles  ADB  and  BDC  be  ad- 
jacent and  in  the  same  plane.  The 
angle  ADC  is  plainly  equal  to  the  sum 
of  the  other  two  (9). 

D 


Problem. — An  angle  mag  he  suhtraded  from  a  greater 


one. 


For  the  angle  ADB  is  the  difference  between  ADO 
and  BDC. 


ANGLES. 


35 


It  IS  equally  evident  that  an  angle  may  be  a  multiple 
or  a  part  of  another  angle ;  in  a  word,  that  angles  are 
quantities  which  may  be  compared,  added,  subtracted, 
multiplied,  or  divided. 

But  angles  are  not  magnitudes,  for  they  have  no  ex- 
tent, either  linear,  superficial,  or  solid. 


ANGLES    FORMED    AT    ONE   POINT. 

88.  Theorem. — The  sum  of  all  the  successive  angles 
formed  in  a  j)lane  upon  one  side  of  a  straight  line,  is  an 
invariable  quaniity;  that  is,  all  such  sums  are  equal  to 
each  other. 

If  AB  and  CD  be  two  straight  lines,  then  the  sum  of  all 
the  successive  angles  at  E  is  equal 
to  the  sum  of  all  those  at  F. 

For  the  line  AE  may  be  placed 
on  CF,  the  point  E  on  the  point 
F.  Then  EB  will  f^ill  on  FD, 
for  when  two  straight  lines  coin- 
cide in  part,  they  must  coincide 
tliroughout  their  mutual  extent 
(51).     Therefore,  the  sum  of  all  ~~ 

the  angles  upon  AB  exactly  coincides  with  the  sum  of 
all  the  angles  upon  CD,  and  the  two  sums  are  equal. 

89.  When  one  line  meets  another, 
making  the  adjacent  angles  equal,  the 
angles  are  called  Rkhit  Angles. 

One  line  is  Perpendicular  to  the 
other  when  the  angle  which  they  make 
is  a  right  angle. 

Two  lines  are  Oblique  to  each  other 
when  they  make  an  angle  which  is  greater  or  less  than 
a  rip^ht  angle. 


36  ELEMENTS    OF  GEOMETRY. 

OO.  Corollary — All  right  angles  are  equal. 

For  each  is  half  of  the  sum  of  the  angles  upon  one 
side  of  a  straight  line.  By  the  above  theorem,  these 
sums  are  always  equal,  and  (7)  the  halves  of  equal 
quantities  are  equal. 

91.  Corollary. — The  sum  of  all  the  successive  angles 
formed  in  a  plane  and  upon  one  side  of  a  straight  line, 
is  equal  to  two  right  angles. 

02.   Corollary — The   sum  of  all 
the   successive   angles  formed   in  a         \^^ 
plane  about  a  point,  is  equal  to  four 
right  angles. 

93.  Corollary — When  two  lines  / 
cut  each  other,  if  one  of  the  angles             / 
thus  formed  is   a  right  angle,  the  other  three  must  be 
right  angles. 

94.  In  estimating  or  measuring  angles  in  geometry, 
the  right  angle  is  taken  as  the  standard. 

An  angle  less  than  a  right  angle  is  called  Acute. 

An  angle  greater  than  one  right  angle  and  less  than 
the  sum  of  two,  is  called  Obtuse.  Angles  greater  than 
the  sum  of  two  right  angles  are  rarely  used  in  ele- 
mentary geometry. 

When  the  sum  of  two  angles  is  equal  to  a  right  angle, 
each  is  the  Complement  of  the  other. 

When  the  sum  of  two  angles  is  equal  to  two  right 
angles,  each  is  the  Supplement  of  the  other. 

95.  Corollary. — Angles  which  are  the  complement  of 
the  same  or  of  equal  angles  are  equal  (7). 

96.  Corollary. — Angles  which  are  the  supplements 
of  the  same  or  of  equal  angles  are  equal. 

97.  Corollary. — The  supplement  of  an  obtuse  angle 
is  acute. 


ANGLES.  37 

08.  Corollary. — The  greater  an  angle,  the  less  is  its 
supplcme.it. 

99.  Corollary — Vertical  angles 
are  equal.  Thus,  a  and  i  are  each 
supplements  of  e. 

100.  Theorem — When  the  sum  of  several  angles  in  a 
plane  having  their  vertices  at  one  point  is  equal  to  two 
right  angles,  the  extreme  sides  form  one  straight  line. 

If  the  sum  of  AGB,  BGC, 
etc.,  be  equal  to  two  right  an- 
gles, then  will  AGF  be  one 
straight  line. 

For  the  sum  of  all  these 
angles  being  equal  (91)  to  the 
sum  of  the  angles  upon  one  side  of  a  straight  line,  it 
follows  that  the  two  sums  may  coincide  (40),  or  that 
AGF  may  coincide  with  a  straight  line.  Therefore, 
AGF  is  a  straight  line. 

EXERCISES. 

101.  Wliich   is  the   greater  angle, 

a  or  ^,  and  why?  ^ — " 

What  is  the  greatest  number  of"  points    ^^""^-'"^^ 
in  which  two   straight    lines  may  cut 
each  other?     In  which  three  may  cut  each  other?     Pour? 

10!2.  The  student  should  ask  and  answer  the  question  "why" 
at  each  step  of  every  demonstration ;  also,  for  every  corollary. 
Thus : 

Why  are  vertical  angles  equal  ?  Why  are  supplements  of  the 
same  angles  equal  ? 

And  in  the  last  theorem:  Why  is  AGF  a  straight  line?  Why 
may  AGF  coincide  with  a  straight  line?  Why  may  the  two  sums 
named  coincide  ?     Why  are  the  two  sums  of  angles  equal  ? 


3S  ELExMENTS   OF   GEOMETKV^. 


PERPENDICULAR   AND   OBLIQUE    LINES. 

103.  Theorem. —  There  can  he  only  one  lijie  through  a 
given  point  perpendicular  to  a  given  straight  line. 

For,  since  all  right  angles  are  equal  (90),  all  lines  ly- 
ing in  one  plane  and  perpendicular  to  a  given  line,  must 
have  the  same  direction.  Now,  through  a  given  point  in 
one  direction  there  can  be  only  one  straight  line  (49). 

Therefore,  since  the  perpendiculars  have  the  same 
direction,  there  can  be  through  a  given  point  only  one 
perpendicular  to  a  given  straight  line. 

When  the  point  is  in  the  given  line,  this  theorem  must 
be  limited  to  one  plane. 

104.  Theorem — If  a  perpendicular  and  oblique  lines 
fall  from  the  same  point  upon  a  given  straight  line,  the 
perpendicular  is  shorter  than  any  oblique  line. 

If  AD  is  perpendicular  and  AC 
oblique  to  BE,  then  AD  is  shorter 
than  AC. 

Let  the  figure  revolve  upon  BE  as  — 
upon  an  axis  (61),  the  point  A  falling 
upon  F,  and  the  lines  AD  and  AC  upon 
FD  and  FC. 

Now,  the  angle  CDF  is  equal  to  the  angle  CD  A,  and 
both  are  right  angles.  Therefore,  the  sum  of  those  two 
angles  being  equal  to  two  right  angles  (100),  ADF  is  a 
straight  line,  and  is  shorter  than  ACF  (54).  There- 
fore, AD,  the  half  of  ADF,  is  shorter  than  AC,  the 
half  of  ACF. 


105.  Corollary. — The  distance  from  a  point  to  a 
straight  line  is  the  perpendicular  let  fall  from  the  point 
to  the  line. 


PERPENDICULAR    AND    OBLIQUE    LINES.  39 

106.  Theorem. — If  a  perpendicular  and  several  oblique 
lines  fall  from  the  same  point  upon  a  given  straight  line, 
and  if  two  oblique  lines  meet  the  given  line  at  equal  dis- 
tances from  the  foot  of  the  perpendicular,  the  two  are 
equal. 

Let  AD  be  the  perpendicular  ^ 

and  AC  and  AE  the  oblique  lines, 
making  CD  equal  to  DE.  Then 
AC  and  AE  are  equal.  

Let  that  portion  of  the  figure  ^  C  D  E  F 
on  the  left  of  AD  turn  upon  AD.  Since  the  angles 
ADB  and  ADF  are  equal,  DB  will  take  the  direction 
DF ;  and  since  DC  and  DE  are  equal,  the  point  C  will 
fall  on  E.  Therefore,  AC  and  AE  will  coincide  (51), 
and  are  equal. 

107.  Corollary — When  the  oblique  lines  are  equal, 
the  angles  which  they  make  with  the  perpendicular  are 
equal.     For  CAD  may  coincide  with  DAE. 

108.  Theorem — Jf  a  line  be  perpendicular  to  another 
at  its  center,  then  every  point  of  the  perpendicular  is 
equally  distant  from  the  two  ends  of  the  other  line. 

For  straight  lines  extending  from  any  point  of  the 
perpendicular  to  the  two  ends  of  the  other  line  must 
be  equal  (106). 

Let  the  student  make  a  diagram  of  this.  Then  state 
what  lines  are  given  by  the  hypothesis,  and  what  are 
constructed  for  demonstration. 

lOO.  Corollary — Since  two  points  fix  the  position 
of  a  line,  if  a  line  have  two  points  each  equidistant  from 
the  ends  of  another  line,  the  two  lines  are  perpendicular 
to  each  other,  and  the  second  line  is  bisected. 

The  two  points  may  be  on  the  same  side,  or  on 
opposite  sides  of  the  second  line. 


40 


ELEMENTS    OF    GKOMETRY. 


no.  Theorem — If  a  perpendicular  and  several  oblique 
lines  fall  from  the  same  point  on  a  given  straigld  line, 
of  two  oblique  lines,  that  which  meets  the  given  line  at  a 
greater  distance  from  the  perpendicular  is  the  longer. 

If  AD  be   perpendicular  to   BG,  and  DF  is  greater 
than    DC,   then   AF    is 
greater  than  AC. 

On  the  line  DF  take 
a  part  DE  equal  to  DC, 
and  join  AE.  Then  let 
the  figure  revolve  upon 
BG,  the  point  A  falling  i    / 

upon   H,  and  the   lines  j  /    / 

AD,  AE,  and  AF  upon  j/X' 

HD,  HE,  and  HF.  H 

Now,  AEH  is  shorter  than  AFH  (79) ;  therefore,  AE, 
the  half  of  AEH,  is  shorter  than  AF,  the  half  of  AFH. 
But  AC  is  equal  to  AE  (106).  Hence,  AF  is  longer 
than  AC,  or  AE,  or  any  line  from  A  meeting  the  given 
line  at  a  less  distance  from  D  than  DF. 

111.  Corollary — A  point  may  be  at  the  same  distance 
from  two  points  of  a  straight  line,  one  on  each  side  of 
the  perpendicular;  but  it  can  not  be  at  the  same  dis- 
tance from  more  than  two  points. 

112.  Theorem  — If  a  line  be  perpendicular  to  another 
at  its  center,  every  point  out  of  the  perpendicular  is  nearer 
to  that  end  of  the  line  which  is  on  the  same  side  of  the 
perpendicular. 

If  BF  is  perpendicular  to  AC 
at  its  center  B,  then  D,  a  point 
not  in  BF,  is  nearer  to  C  than 
to  A. 

Join  DA  and  DC,  and  let  the       A  BE 

perpendicular  DE  fall  from  D  upon  the  line  AC. 


PERPENDICULAR    AND    OBLIQUE    LINES.  41 

This  perpendicular  must  fall  on  the  same  side  of  BF 
as  the  point  D,  for  if  it  crossed  the  line  BF,  there  would 
be  from  the  point  of  intersection  two  perpendiculars  on 
AC,  which  is  impossible  (103).  Now,  since  AB  is  equal 
to  BC,  AE  must  be  greater  than  EC.  Hence,  AD  is 
greater  than  CD  (110). 

The  point  D  is  supposed  to  be  in  the  plane  of  ACF. 
If  it  were  not,  the  perpendicular  from  it  might  fall  on 
the  point  B. 

BISECTED    ANGLE. 

113.  Theorem — Every  point  of  the  line  which  bisects 
an  angle  is  equidistant  from  the  sides  of  the  angle. 

Let  BCD  be   the   given   angle,  / 

and  AC  the  bisecting  line.     Then   ^  / 

the  distance  of  the  two  sides  from      \  //j 

any  point  A  of  that  line  is  meas-  \^'''   /  j 

ured    by    perpendiculars    to    the  \Z_J 

sides,  as  AF  and  AE.  C     E  D 

Since  the  angles  BCA  and  DCA  are  equal,  that  part 
of  the  figure  upon  the  one  side  of  AC  may  revolve  upon 
AC,  and  the  line  BC  will  take  the  direction  of  CD,  and 
coincide  with  it. 

Then  the  perpendiculars  AF  and  AE  must  coincide 
(103),  and  the  point  F  fall  upon  E.  Therefore,  AF  and 
AE  are  equal,  and  the  point  A  is  equally  distant  (105) 
from  the  sides  of  the  given  angle. 


APPLICATION. 

114.  Perpendicular  lines  are  constantly  used  in  architecture, 
carpentry,  stone-cutting,  nmcliinery,  etc. 

I'he  mason's  square  consists  of  two  flat  rulers  made  of  iron, 
and  connected  togetlier  in  such  a  manner  that  both  edges  of  one 
Geoni. — 4 


42 


ELEMENTS    OF    GEOMETUV 


are  at  right  angles  to  those  of  the  otlier.     The  carpenter's  square 
is  much  like  it,  but  one  of  the  legs  is 
wood.     This   instrument  is  used    for 
drawing  perpendicular  lines,  and  for 
testing  the  correctness  of  right  angles. 

The  square  itself  should  be  tested 
in  the  following  manner: 

On  any  plane  surface  draw  an  angle,  as  BAG,  with  the  square. 
Extend  BA  in  the  same  straight  line 
to  D.  Then  turn  the  square  so  that 
the  edges  by  which  the  angle  BAG 
was  described,  may  be  applied  to  the 
angle  DAG.  If  the  coincidence  is 
exact,  the  square  is  correct  as  to 
these  edges. 

Let  the  student  show  that  this  method  of  testing  the  square  is 
according  to  geometrical  principles. 

The  square  here  described  is  not  the  geometrical  figure  of  that 
name,  which  will  be  defined  hereafter. 


B 


A    MINIMUM    LINE 


115.  Theorem. — Of  any  two  lines  wTiicJi  may  extend 
from  hvo  given  points  outside  of  a  straight  line  to  any 
point  iyi  it,  those  which  are  together  least  make  equal  an- 
gles with  that  line. 

Let  CD  be  the  line  and  A  and  B  the  points,  and 
AEB  the  shortest  line  that 
can  be  made  from  A  to  B 
through  any  point  of  CD. 
Then  it  is  to  be  proved 
that  AEC  and  BED  are 
etjual  angles. 

Make  AH  perpendicular 
to  CD,  and  produce  it  to  F,  making  HF  equal  to  AH. 

Now  every  point  of  the  line  CD  is  equally  distant 
from  A  and  F  (108).     Therefore,  every  line  joining  B  to 


PARALLELS.  43 

F  through  some  point  of  CD,  is  equal  to  a  line  joining 
B  to  A  through  the  same  point.  Thus,  BGF  is  equal  to 
BGA,  since  GF  and  GA  are  equal.  So,  BEF  is  equal 
to  BEA. 

But  BEA  is,  by  hypothesis,  the  shortest  line  from  B 
to  A  through  any  point  of  CD.  Therefore,  BEF  is  the 
shortest  line  from   B  to  F,  and  is  a  straight  line  (54). 

Since  BEF  is  one  straight  line,  the  angles  FEH  and 
BED  are  vertical  and  equal  (99).  But  the  angles  FEH 
and  AEH  are  equal  (107).  Therefore,  AEH  and  BED 
are  equal  (6). 

116.  When  several  magnitudes  are  of  the  same  kind 
but  vary  in  extent,  the  least  is  called  a  minimum^  and 
the  greatest  a  maximum, 

APPLICATION. 

When  a  ray  of  light  is  reflected  from  a  polished  surface,  the 
incident  and  reflected  parts  of  the  ray  make  equal  angles  with 
the  surface.  We  learn  from  this  geometrical  principle  that  light, 
when  reflected,  still  adheres  to  that  law  of  its  nature  which  re- 
quires it  to  take  the  shortest  path. 

PARALLELS. 

IIT.  Parallel  lines  are  straight  lines  which  have 
the  same  directions. 

118.  Corollary — Two  lines  which  are  each  parallel  to 

a  third  are  parallel  to  each  other. 

119.  Corollary — From  the  above  definition,  and  the 
Axiom  of  Direction  (49),  it  follows  that  there  can  be  only 
one  line  through  a  given  point  parallel  to  a  given  line. 

120.  Corollary — From  the  same  premises,  it  follows 
that  two  parallel  lines  can  never  meet,  or  have  a  com- 
mon point. 


44  ELEiVlEiM'«    OF    GEOMETRY. 

\2\»  Theorem — Two  parallel  lines  both  lie  in  one 
plane  and  determine  its  position. 

The  position  of  a  plane  is  determined  (60)  by  either 
line  and  one  point  of  the  other  line.  Now  the  plane 
has  the  direction  of  the  first  line  and  can  not  vary  from 
it  (56),  and  the  second  line  has  also  the  same  direction 
(117)  and   can   not  vary  from   it  (44). 

Therefore,  the  second  line  must  also  lie  wholly  in  the 
plane. 

NAMES    OF    ANGLES. 

122.  When  two  straight  lines  are  cut  by  a  secant, 
the  eight  angles  thus  formed  are 
named  as  follows: 

The  four  angles  between  the 
two  lines  are  Interior;  as,/, ^, 
h,  and  k.  The  other  four  are  Ex- 
terior; as,  b,c,  I,  and  m. 

Two  angles  on  the  same  side  of  the  secant,  and  on 
the  same  side  of  the  two  lines  cut  by  it,  are  called  Cor- 
responding angles.  The  angles  h  and  b  are  corre- 
sponding. 

Two  angles  on  opposite  sides  of  the  secant,  and  on 
opposite  sides  of  the  two  lines  cut  by  it,  are  called 
Alternate  angles.  The  angles  /  and  k  are  alternate  ; 
also,  b  and  m. 

The  student  should  name  the  corresponding  and  the 
alternate  angles  of  each  of  the  eight  angles  in  the  above 
diagram.  Let  him  also  name  them  in  the  diagram  of 
the  following  theorem. 

12]$.  Corollary — The  corresponding  and  the  altern- 
ate angles  of  any  given  angle  are  vertical  to  each  other, 
and  therefore  equal  (99). 


PARALLELS.  45 


PARALLELS    CUT    BY  A    SECANT. 

134.  Theorem. —  When  two  parallel  lines  are  cut  hy  a 
secant^  each  of  the  eight  angles  is  equal  to  its  corresponding 
angle. 

If  the  straight  lines  AB   and  CD  have  the  sam6  di- 
rections, then  the  angles  FHB 
and  FGD  are  equal. 

For,  since  the  directions 
GD  and  HB  are  the  same,  the 
direction  GF  diifers  equally 
from  them.  Therefore,  the 
angles  are  equal  (82). 

In  the  same  manner,  it  may 
be  shown  that  any  two  corresponding  angles  are  equal. 

125.  Corollary — When  two  parallel  lines  are  cut  by 
a  secant,  each  of  the  eight  angles  is  equal  to  its  altern- 
ate (128). 

126.  Corollary — Two  interior  angles  on  the  same 
side  of  the  secant  are  supplements  of  each  other.  For, 
since  GHB  is  the  supplement  of  FHB  (91),  it  is  also 
the  supplement  of  its  equal  HGD.  Two  exterior  angles 
on  the  same  side  of  the  secant  are  supplementary,  for 
a  similar  reason. 

127.  Corollary — When  a  secant  is  perpendicular  to 
one  of  two  parallels,  it  is  also  perpendicular  to  the  other, 
and  all  the  angles  are  right. 

Let  the  student  illustrate  by  a  diagram,  in  this  and 
in  all  cases  when  a  diagram  is  not  given. 

128.  Corollary — When  the  secant  is  oblique  to  the 
parallels,  four  of  the  angles  formed  are  obtuse  and  are 
equal  to  each  other ;  the  other  four  are  acute,  and  equal ; 
und  any  acute  angle  is  the  supplement  of  any  obtuse. 


4«  ELEMENTS    OF    GEOMETRY. 

ISO.  Theorem. —  When  two  straight  lines,  being  in  the 
same  plane,  are  cut  hy  a  third,  making  the  corresponding 
angles  equal,  the  two  lines  so  cut  are  parallel. 

If  AB  and  CD  lie  in  the  same  plane,  and  if  the  angles 
AHF    and    CGF   are    equal, 
then  AB  and  CD  are  parallel. 

For,  suppose  a  straight  line 
to  pass  through  the  point  H, 
paralle]  to  DC.  Such  a  line 
makes  a  corresponding  angle 
equal  to  CGF,  and  therefore 
equal  to  AHF.  This  sup- 
posed parallel  line  lies  in  the  same  plane  as  CD  and 
H  (121);  that  is,  by  hypothesis,  in  the  same  plane  as 
AB.  But  if  it  lies  in  the  same  plane  with  AB  and 
makes  the  same  angle  with  the  same  line  EF,  at  the 
same  point  H,  then  it  must  coincide  with  AB.  For, 
when  two  angles  are  equal  and  placed  one  upon  the 
other,  they  coincide  throughout.  Therefore,  AB  is  par- 
allel to  CD. 

130.  Corollary — If  the  alternate  angles  are  equal, 
the  lines  are  parallel  (123). 

131.  Corollary — The  same  conclusion  must  follow 
when  the  interior  angles  on  the  same  side  of  the  secant 
are  supplementary. 

DISTANCE    BETWEEN    PARALLELS. 

13!S.  Theorem Two  parallel   lines   are    everywhere 

equally  distant. 

The  distance  between  two  parallel  lines  is  measured 
by  a  line  perpendicular  to  them,  since  it  is  the  short- 
est from  one  to  the  other. 

Let  AB   and   CD  be   two    parallels.     Then  any  per- 


PARALLELS. 


47 


A     E 


G     B 


C     F 


M 


H     D 


peiidiculars  to  them,  as  EF  and  GH,  are  equal.     From 
M,  the  center  of  FH,  erect  the  perpendicular  ML. 

Let  that  part  of  the  figure 
to  the  left  of  ML  revolve 
upon  ML.  All  the  angles 
of  the  figure  being  right 
angles,  MC  will  fall  upon 
MD.  Since  MF  is  equal  to  MH,  the  point  F  will  fall 
on  H,  and  the  angles  at  F  and  H  being  equal,  FE  will 
take  the  direction  HG,  and  the  point  E  will  be  on  the 
line  HG.  But  since  the  angles  at  L  are  equal,  the 
point  E  will  also  fall  on  LB,  and  being  on  both  LB  and 
HG,  it  must  be  on  G.  Therefore,  FE  and  HG  coincide 
and  are  equal. 

133.  Corollary — The  parts  of  parallel  lines  included 
between  perpendiculars  to  them,  must  be  equal.  For 
the  perpendiculars  are  parallel  (129). 


SECANT    AND     PARALLELS. 

134.  Theorem — If  several  equally  distant  parallel  lines 
he  cut  by  a  secant,  the  secant  will  he  divided  into  equal 
parts. 

If  the  parallels  BC,  DF,  GH,  and  KL  are  at  equal 
distances,  then  the  parts  ^ 

EI,  10,  and  OU  of  the 
secant  AY  are  equal. 

For  that  part  of  the 
figure  included  between 
BC  and  DF  may  be 
placed  upon  and  will 
coincide  with  that  part 
between  DF  and  GH ; 
for  the  parallels  are  everywhere  equally  distant  (132). 


B 

\e 

c 

D 

\l 

F 

G 

\o 

H 

K 

v 

L 

48  ELEMENTS   OF   GEOMETRY. 

Let  them  be  so  placed  that  the  point  E  may  fall  upon  I. 
Then,  since  the  angles  BEI  and  DIO  (124)  are  equal, 
the  line  EI  will  take  the  direction  10.  And  since  DF 
and  GH  coincide,  the  point  I  will  fall  on  0.  Therefore, 
EI  and  10  coincide  and  are  equal.  In  like  manner, 
show  that  any  two  of  the  intercepted  parts  of  the  line 
AY  are  equal. 

135.  Corollary. — Conversely,  if  several  parallel  lines 
intercept  equal  segments  of  a  secant,  then  the  several 
distances  between  the  parallels  are  equal. 

136.  Corollary. — When  the  distances  between  the 
parallels  are  unequal,  the  segments  of  the  secant  are 
unequal.  And  conversely,  when  the  segments  of  the 
secant  are  unequal,  the  distances  are  unequal. 

LINES    NOT    PAEALLEL    MEET. 

137.  Theorem. — If  two  straight  lines  are  in  the  same 
plane  and  are  not  parallel^  they  will  meet  if  sufficiently 
produced. 

Let   AB    and    CD    be  two   lines.     Let  the  line  EF, 
parallel   to  CD,   pass 
through  any  point  of                    _^^^ 
AB,  as  H.     From  H        E- .:::::::::^[:^^     F 

let  the  perpendicular — ^ 

HG  foil  upon  CD.  J 

Since  AB  and  EF       ^ \^  ^ 

have  different  direc- 
tions, they  cut  each  other  at  the  point  H.  Take  any 
point,  as  I,  in  that  part  of  AB  which  lies  between  EF 
and  CD,  and  extend  a  line  IK  parallel  to  CD  through 
the  plhnt  I.  Now  divide  HG  into  parts  equal  to  HK 
until  one  of  the  points  of  division  falls  beyond  G. 
Then  along   IIB,  take  parts  equal   to  HI,  as  often  as 


PARALLELS. 


49 


HK  was  taken  along  HG.  Lastly,  from  each  point  of 
division  of  HB,  extend  a  line  perpendicular  to  HG. 

These  perpendiculars  are  parallel  to  each  other  and 
to  CD  (129).  These  parallels  by  construction  intercept 
equal  parts  of  HB.  Therefore  (135),  they  are  equally 
distant  from  each  other.  Hence,  HG  is  divided  by 
them  into  equal  segments  (134);  that  is,  each  one 
passes  through  one  of  the  previously  ascertained  points 
of  the  line  HG. 

But  the  last  of  these  points  was  beyond  the  line  CD, 
and  as  the  parallel  can  not  cross  CD  (120),  the  corre- 
cponding  point  of  HB  is  beyond  CD.  Therefore,  HB 
and  CD  must  cross  each  other. 


ANGLES   WITH    PARALLEL    SIDES. 

138.  Theorem. —  When  the  sides  of  one  angle  are  par- 
allel to  the  sides  of  another^  and  have  respectively  the  same 
directions  from  their  vertices^  the  two  angles  are  equal. 

If  the  directions  BA  and  DC  are  the  same,  and  the 
directions  DE  and  BF  are  the 
same,  then  the  angles  ABF  and 
CDE  are  equal. 

For  each  of  these  angles  ig 
equal  to  the  angle  CGF  (124). 

139.  Let  the  student  dem- 
onstrate that  when  two  of  the 
parallel  sides  have  opposite  di- 
rections, and  the  other  two  have 
the  same  direction,  then  the 
angles  are  supplementary. 

Let  him  also  demonstrate  that  if  both  sides  of  one 
angle  have  directions  respectively  opposite  to  those  of 
the  other,  then  again  the  angles  are  equal. 
(Teoni. — 5 


50 


ELEMENTS   OF   GEOMETRY. 


ANGLES   WITH    PERPENDICULAR    SIDES. 

140.  Theorem — Two  angles  which  have  their  sides  re- 
spectively perpendicular  are  equal  or  supplementary. 

If  AB  is  perpendicular  to  DG,  and  BC  is  perpendicu- 
lar to  EF,  then  the 
angle  ABC  is  equal  to 
one,  and  supplement- 
ary to  the  other  of  the  i 
angles  formed  by  DG 
and  EF  (86). 

Through  B  extend 
BI  parallel  to  GD,  and 
BH  parallel  to  EF. 

Now,  ABI  and  CBH 
are  right  angles  (127),  and  therefore  equal  (90).  Sub- 
tracting the  angle  HBA  from  each,  the  remainders  HBI 
and  ABC  are  equal  (7).  But  HBI  is  equal  to  FGD 
(138),  and  is  the  supplement  of  EGD  (139).  Therefore, 
the  angle  ABC  is  equal  or  supplementary  to  any  angle 
formed  by  the  lines  DG  and  EF. 


APPLICATIONS 


141.  The  instrument  called  the  T  square  consists  of  two  straight 

and  flat  rulers  fixed  at  ri";ht  angles  to  each 

other,  as  in  the  figure.     It  is  used  to  draw 
parallel  lines. 

Draw  a  straight  line  in  a  direction  per- 
pendicular to  that  in  which  it  is  required  to 
draw  parallel  lines.  Lay  the  cross-piece  of 
(he  T  ruler  along  this  line.  The  other 
piece  of  the  ruler  gives  the  direction  of  one 
of  the  parallels.  The  ruler  being  moved  along  the  paper,  keep' 
ing  the  cross-piece  coincident  with  the  line  first  described,  any 
number  of  parallel  lines  may  be  drawn. 


PARALLELS.  51 

What  is  the  principle  of  geometry  involved  in  the  use  of  this 
instrument? 

142*  The  uniform  distance  of  parallel  lines  is  the  principle 
upon  which  numerous  instruments  and  processes  in  the  arts  are 
founded. 

If  two  systems,  each  consisting  of  several  parallel  lines,  cross 
each  other  at  right  angles,  all  the  parts  of  one  system  included 
between  any  two  lines  of  the  other  system  will  be  equal.  Tlie 
ordinary  framing  of  a  window  consists  of  two  systems  of  lines  of 
this  kind;  the  shelves  and  upright  standards  of  book-cases  and 
the  paneling  of  doors  also  afford  similar  examples. 

143.  The  joiner's  gauge  is  a  tool  with  which  a  line  may  be 
drawn  on  a  board  parallel  to  its  edge.  It  consists  of  a  square 
piece  of  wood,  with  a  sharp  steel  point  near  the  end  of  one  wide, 
and  a  movable  band,  which  may  be  fastened  by  a  screw  or  key  at 
any  required  distance  from  the  point.  The  gauge  is  held  perpen- 
dicular to  the  edge  of  the  board,  against  which  the  band  is 
pressed  while  the  tool  is  moved  along  the  board,  the  steel  point 
tracing  the  parallel  line. 

144.  It  is  frequently  important  in  machinery  that  a  body  shall 
have  what  is  called  a  parallel  motion  ;  that  is,  such  that  all  its  parts 
shall  move  in  parallel  lines,  preserving  the  same  relative  position 
to  each  other. 

The  piston  of  a  steam-engine,  and  the  rod  which  it  drives,  re- 
ceive such  a  motion ;  and  any  deviation  from  it  would  be  attended 
with  consequences  injurious  to  the  machinery.  The  whole  mass 
of  the  piston  and  its  rod  must  be  so  moved,  that  every  point  of 
it  shall  describe  a  line  exactly  parallel  to  the  direction  -^f  the 
cylinder. 


52 


ELEMENTS    OF   GEOMETRY. 


CHAPTER    IV. 


THE    CIRCUMFERENCE. 


145.  Let  the  line  AB  revolve  in  a  plane  about  the 
^nd  A,  which  is  fixed.  Then  the 
point  B  will  describe  a  line  which 
returns  upon  itself,  called  a  cir- 
cumference of  a  circle.  Hence, 
the  following  definitions : 

A   Circle   is   a   portion   of  a       .  ^^ 

plane  bounded   by  a  line   called 
a  Circumference,  every  point  of 

which   is  equally  distant  from  a  point  Avithin  called  the 
Center. 

146.  Theorem. — A  circumference  is  curved  throughout. 
For  a  straight  line  can  not  have  more  than  two  points 

equally  distant  from  a  given  point  (111). 

14T.  Corollary. — A  straight  line  can  not  cut  a  cir- 
cumference in  more  than  two  points. 

148.  The  circumference  is  the  only  curve  considered 
in  elementary  geometry.  Let  us  examine  the  proper- 
ties of  this  line,  and  of  the  straight  lines  which  may  be 
combined  with  it. 


HOW    DETERMINED. 


149.  Theorem. —  Three  points  7iot  in  the  same  straight 
line  fix  a  circumference  both  as  to  position  and  extent. 
The   three   given   points,  as  A,  B,  and   C,  determine 


ARCS    AND   RADII.  53 

the  position  of  a  plane.  Let  the  given  points  be  joined 
by  straight  lines  AB  and 

A  D  "R 

BC.    At  D  and  E,  the  mid-      — -y \e 

die  points  of  these  lines,  let  •  /  \C 

perpendiculars  be  erected  i 

in  the  plane  of  the  three  q: 

points.  ;        /jj 

By  the  hypothesis,  AB  |     / 

and  BC  make  an  angle  at  !  / 

B.     Therefore,  GD  is  not  Jl 

perpendicular  to   BC,  for  /  i 

if  it  were,  AB  and  BC  would  be  parallel  (129).  Hence, 
DG  and  EH  are  not  parallel  (117),  since  one  is  per- 
pendicular and  the  other  is  not  perpendicular  to  BC. 
Therefore,  DG  and  EH  will  meet  (137)  if  produced. 
Let  L  be  their  point  of  intersection. 

Since  every  point  of  DG  is  equidistant  from  A  and  B 
(108),  and  since  every  point  of  EH  is  equidistant  from 
B  and  C,  their  common  point  L  is  equidistant  from  A, 
B,  and  C.  Therefore,  with  this  point  as  a  center,  a 
circumference  may  be  described  through  A,  B,  and  C. 
There  can  be  no  other  circumference  through  these 
three  points,  for  there  is  no  other  point  besides  L 
equally  distant  from  all  three  (112). 

Therefore,  these  three  points  fix  the  position  and  the 
extent  of  the  circumference  which  passes  through  them. 

ARCS    AND    RADII. 

150.  An  Arc  is  a  portion  of  a  circumference. 

A  Radius  is  a  straight  line  from  the  center  to  the 
circumference. 

A  Diameter  is  a  straight  line  passing  through  the 
center,  and  limited  at  both  ends  by  the  circumference. 

A  Chord  is  ?  straight  line  joining  the  ends  of  an  arc. 


54  ELEMENTS    OF    GEOMETRY. 

151.  Corollary. — All  radii  of  the  same  circumference 
are  equal. 

15!S.  Corollary — In  the  same  circumference,  a  diame- 
ter is  double  the  radius,  and  all  diameters  are  equal. 

153.  Corollary. — Every  point  of  the  plane  at  greater 
distance  from  the  center  than  the  length  of  the  radius, 
is  outside  of  the  circumference.  Every  point  at  a  less 
distance  from  the  center,  is  within  the  circumference. 
Every  point  whose  distance  from  the  center  is  equal  to 
the  radius,  is  on  the  circumference. 

154.  Theorem. — CircumferenccB  ivTiich  have  equal  radii 
are  equal. 

Let  the  center  of  one  be  placed  on  that  of  the  other. 
Then  the  circumferences  will  coincide.  For  if  it  were 
otherwise,  then  some  points  would  be  unequally  distant 
from  the  common  center,  which  is  impossible  when 
the  radii  are  equal.  Therefore,  the  circumferences  are 
equal. 

155.  Corollary — A  circumference  may  revolve  upon, 
or  slide  along  its  equal. 

156.  Corollary. — Two  arcs  of  the  same  or  of  equal 
circles  may  coincide  so  far  as  both  extend. 

157.  Theorem. — Every  diameter  bisects  the  circumfer 
ence  and  the  circle. 

For  that  part  upon  one  side  of  the  diameter  may  be 
turned  upon  that  line  as  its  axis.  When  the  two  parts 
thus  meet,  they  will  coincide;  for  if  they  did  not,  some 
points  of  the  circumference  would  be  unequally  distant 
from  the  center. 

158.  A  line  which  divides  any  figure  in  this  manner, 
is  said  to  divide  it  syr)imetricalhj ;  and  a  figure  which  can 
be  so  divided  is  symmetrical. 


ARCS    AND    RADII.  55 

159.  Theorem — A  diameter  is  greater  than  any  other 
fihord  of  the  same  circumference. 

To  be  demonstrated  by  the  student. 

160.  Problem Arcs  of  equal  radii  may  he  added  to- 
gether, or  one  may  he  suhtracted  from  another. 

For  an  arc  may  be  produced  till  it  becomes  an  entire 
circumference,  or  it  may  be  diminished  at  will  (35  and 
145). 

Therefore,  the  length  of  an  arc  may  be  increased  or 
decreased  by  the  length  of  another  arc  of  the  same  ra- 
dius; and  the  result,  that  is,  the  sum  or  diiference,  will 
be  an  arc  of  the  same  radius. 

161.  Corollary — Arcs  of  equal  radii  may  be  multi- 
plied or  divided  in  the  same  manner  as  straight  lines. 

163.  The  sum  of  several  arcs  may  be  greater  than 
a  circumference. 

163.  Two  arcs  not  having  the  same  radius  may  be 
joined  together,  and  the  result  may  be  called  their  sum ; 
but  it  is  not  one  arc,  for  it  is  not  a  part  of  one  circum- 
ference. 

APPLICATIONS. 

164.  The  circumference  is  the  only  line  which  can  move  along 
itself,  around  a  ct.iter,  vvitliout  suffering  any  change.  For  any 
line  that  can  do  this  must,  therefore,  have  all  its  points  equally 
distant  from  the  center  of  revolution ;  that  is,  it  must  be  a  cir- 
cumference. 

It  is  in  virtue  of  this  property  that  the  axles  of  wheels,  shafts, 
and  other  solid  bodies  which  are  required  to  revolve  witlun  a  hol- 
low mold  or  casing  of  their  own  form,  must  be  circular.  If  they 
were  of  any  other  form,  they  would  be  incapable  ot  revolving  with- 
out carrying  the  mold  or  casing  around  with  them. 

165.  Wheels  which  are  intended  to  maintain  a  carriage  always 
at  the  same  hight  above  the  road  on  which  they  rol),  must  be  cir- 
cular, with  the  axle  in  the  center. 


aO  ELEMENTS   OF    GEUMETKV. 

166.  The  art  of  turning  consists  in  the  production  of  the  cir- 
cular form  by  mechanical  means.  The  substance  to  be  turned  is 
placed  in  a  machine  called  a  lathe,  which  gives  it  a  rotary  mo- 
tion. The  edge  of  a  cutting  tool  is  placed  at  a  distance  from  the 
axis  of  revolution  equal  to  the  radius  of  the  intended  circle.  As 
the  substance  revolves,  the  tool  removes  every  part  that  is  further 
from  the  axis  than  the  radius,  and  thus  gives  a  circular  form  to 
what  remains. 


PROBLEMS    IN    DRAWING. 

IGT.  The  compasses  enable  us  to  draw  a  circumference,  or  an 
arc  of  a  given  radius  and  given  center. 

Open  the  instrument  till  the  points  are  on  the  two  ends  of  the 
given  radius.  Then  fix  one  point  on  the  given  center,  and  the 
other  point  may  be  made  to  revolve  around  in  contact  with  the 
surface,  thus  tracing  out  the  circumference. 

The  revolving  leg  may  have  a  pen  or  pencil  at  the  point.  In 
the  operation,  care  should  be  taken  not  to  vary  the  opening  of  the 
compasses. 

168.  It  is  evident  that  with  the  ruler  and  compasses  (69), 

1.  A  straight  line  can  be  drawn  through  two  given  points. 

2.  A  given  straight  line  can  be  produced  any  length. 

3.  A  circumference  can  be  described  from  any  center,  with  any 
radius. 

169.  The  foregoing  are  the  three  postulates  of  Euclid.  Since 
the  straight  line  and  the  circumference  are  the  only  lines  treated 
of  in  elementary  geometry,  these  Euclidian  postulates  are  a  sui- 
ficient  basis  for  all  problems.  Hence,  the  rule  that  no  instruments 
shall  be  used  except  the  ruler  and  the  compasses  (68). 

ITO.  In  the  Elements  of  Euclid,  which,  for  many  ages,  was  the 
only  text-book  on  elementary  geometry,  the  problems  in  drawing 
occupy  the  place  of  problems  in  geometry.  At  present,  the  mathe- 
maticians of  Germany,  France,  and  America  put  them  aside  as 
not  forming  a  necessary  part  of  the  theory  of  the  science.  English 
writers,  however,  generally  adhere  to  Euclid 

171.  Problem — To  bisect  a  given  straight  line. 

With  A  and  B  as  centers,  and  with  a  radius  greater  than  the 
half  of  AB,  describe  arcs   which  intersect  in  the   two  points  D 


PROBLEMS    IN    DRAWING.  57 

and  E.     The  straight  line  joining  these  two  points  will  bisect  AB 
at  C. 

Let  the  demonstration  be  given  T) 

by  the  student  (109  and  151).  /\ 

172.  Problem.— To  erect  a 

perpendicular  on  a  given 
straight  line  at  a  given  point. 

Take  two  points  in  the  line, 
one  on  each  side  of  the  given 
point,  at  equal  distances  from  it.  yC 

Describe  arcs  as  in  the  last  prob- 
lem,  and  their  intersection  gives  one  point  of  the  perpendicular. 

Demonstration  to  be  given  by  the  student. 

ITS.  Problem — To  let  fall  a  perpendicular  from  a 
given  point  on  a  given  straight  line. 

With  the  given  point  as  a  cen-  ^ 

ter,  and    a   radius   long   enough,  j 

describe  an  arc  cutting  the  given  j 

line  BC  in  the   points   D  and  E.  j 

The    line    may    be    produced,    if  j 

necessary,  to  be  cut  by  the  arc  in 
two  places.  With  D  and  E  as 
centers,  and  with  a  radius  greater 
than  the  half  of  DE,  describe 
arcs  cutting  each  other  in  F.  The 
straight  line  joining  A  and  F  is  perpendicular  to  DE. 

Let  the  student  show  why. 

174.  Problem — To  draw  a  line  through  a  given  point 
parallel  to  a  given  line. 

Let  a  perpendicular  fall  from  the  point  on  the  line.  Then,  at 
the  given  point,  erect  a  perpendicular  to  this  last.  It  will  be  par- 
allel to  the  given  line. 

Let  the  student  explain  why  (129). 

175.  Problem. —  To  describe  a  circumference  through 
three  given  points. 

The  solution  of  this  problem  is  evident,  from  Article  149. 


Xf 


58 


ELEMENTS    OF    GEOMETRY. 


176,  Problem — To  find  the  center  of  a  given  arc  or 
circumference. 

Take  any  three  points  of  tlie  arc,  and  proceed  as  in  the  last 
problem. 

XKH ,  Tlie  student  is  advised  to  make  a  drawing  of  every  prob- 
lem. First  draw  the  parts  given,  then  the  construction  requisite 
I'or  solution.      Afterward  demonstrate  its  correctness. 

Endeavor  to  make  the  drawing  as  exact  as  possible.  Let  the 
lines  be  fine  and  even,  as  they  better  represent  the  abstract  lines 
of  geometry.  A  geometrical  principle  is  more  easily  understood 
by  the  student,  when  he  makes  a  neat  diagram,  than  when  his 
drawino;  is  careless. 


TANGENT. 

178.  Theorem — A  straight  line  which  is  perpendicular 
to  a  radius  at  its  extremity,  touches  the  circumference  in 
only  one  point. 

Let  AD  be   perpendicular   to   the   radius   BC   at   its 
extremity  B.     Then  it  is  to  be 
proved    that    AD  touches    the 
circumference  at  B,  and  at  no 
other  point. 

If  the  center  C  be  joined  by 
straight  lines  with  any  points 
of  AD,  the  perpendicular  BC 
will  be  shorter  than  any  such 
oblique  line  (104).  Therefore 
(153),  every  point  of  the  line 
AD,  except  B,  is  outside  of 
the  circumference. 

17^.  A  Tangent  is  a  line  touching  a  circumference 
in  only  one  point.  The  circumference  is  also  said  to  be 
tangent  to  the  straight  line.  The  common  point  is 
called  the  point  of  contact. 


SECANT. 


59 


APPLICATION. 

180.  Tangent  lines  are  frequently  used  in  the  arts.  A  com- 
mon example  is  when  a  strap  is  carried  round  a  part  of  the  cir- 
cumference of  a  wheel,  and  extending  to  a  distance,  sufficient 
tension  is  given  to  it  to  produce  such  a  degree  of  friction  between 
it  and  the  wheel,  that  one  can  not  move  without  the  other. 

181.  Problem  in  Drawing. — To  draiv  a  tangent  at  a 
given  point  of  an  arc. 

Draw  a  radius  to  the  given  point,  and  erect  a  perpendicular  to 
the  radius  at  that  point. 

It  will  be  necessary  to  produce  the  radius  beyond  the  arc,  as 
the  student  has  not  yet  learned  to  erect  a  perpendicular  at  the 
e.vtremity  of  a  line  without  producing  it. 


SECANT. 

182.  Theorem. — A  straight  line  which  is  oblique  to  a 
radius  at  its  extremity,  cuts  the  circumference  in  two  points. 

Let  AD  be  oblique  to  the  radius  CB  at  its  extrem- 
ity B.     Then  it  will  cut  the  cir- 
cumference at   B,  and   at  some 
other  point. 

From  the  center  C,  let  CE  fall 
perpendicularly  on  AD.  On  ED, 
take  EF  equal  to  EB. 

Then  the  distance  from  C  to 
any  point  of  the  line  AD  be- 
tween B  and  F  is  less  than  the 
length  of  the  radius  CB  (110), 
and  to  any  point  of  the  line  be- 
yond B  and  F,  it  is  greater  than 
the  length  of  CB.  Therefore  (153),  that  portion  of  the 
line  AD  between  B  and  F  is  within,  and  the  parts  be- 
yond B  and  F  are  without  the  circumference.  Hence, 
the  oblique  line  cuts  the  circumference  in  two  points. 


60  ELEMENTS   OF   GEOxMETRY. 

183.  Corollary. — A  tangent  to  the  circumference  is 
perpendicular  to  the  radius  which  extends  to  the  point 
of  contact.  For,  if  it  were  not  perpendicular,  it  would 
be  a  secant. 

184.  Corollary. — At  one  point  of  a  circumference, 
there  can  be  only  one  tangent  (103). 

CHORDS. 

185.  Theorem The  radii  being  equal,  if  two  arcs  are 

equal  their  chords  are  also  equal. 

If  the  arcs  AGE  and  BCD  are  equal,  and  their  radii 
are  equal,  then  AE  and  BD  are  equal. 


For,  since  the  radii  are  equal,  the  circumferences  are 
equal  (154) ;  and  the  arcs  may  be  placed  one  upon  the 
other,  and  will  coincide,  so  that  A  will  be  upon  B,  and  E 
upon  D.  Then  the  two  chords,  being  straight  lines, 
must  coincide  (51),  and  are  equal. 

180.  Every  chord  subtends  two  arcs,  which  together 
form  the  whole  circumference.  Thus  thp  chord  AE  sub- 
tends the  arcs   AOE  and  AIE. 

The  arc  of  a  chord  always  means  the  smaller  of  the 
two,  unless  otherwise  expressed. 

187.  Theorem. —  The  radius  tvhich  is  perpendicular  to 
a  chord  bisects  the  chord  and  its  arc. 


CHORDS. 


61 


Let  CD  be  perpendicular  at  E  to  the  chord  AB,  then 
will  AE  be  equal  to  EB,  and  the  arc  AD  to  the  arc  DB. 

Produce  DC  to  the  circum- 
ference at  F,  and  let  that  part 
of  the  figure  on  one  side  of 
DF  be  turned  upon  DF  as  upon 
an  axis.  Then  the  semi-circum- 
ference DAF  will  coincide  with 
DBF  (157).  Since  the  angles 
at  E  are  right,  the  line  EA  will 
take  the   direction  of  EB,  and 

the  point  A  will  fall  on  the  point  B.  Therefore,  EA 
and  EB  will  coincide,  and  are  equal;  and  the  same  is 
true  of  DA  and  DB,  and  of  FA  and  FB. 

188.  Corollary — Since  two  conditions  determine  the 
position  of  a  straight  line  (52),  if  it  has  any  two  of  the 
four  conditions  mentioned  in  the  theorem,  it  must  have 
the  other  two.     These  four  conditions  are, 

1.  The  line  passes  through  the  center  of  the  circle, 
that  is,  it  is  a  radius. 

2.  It  passes  through  the  center  of  the  chord. 

3.  It  passes  through  the  center  of  the  arc. 

4.  It  is  perpendicular  to  the  chord. 

189.  Theorem. —  The  radii  being  equals  when  two  arcs 
are  each  less  than  a  semi-circumference,  the  greater  arc  has 
the  greater  chord. 

If  the  arc  AMB  is  greater  than  CND,  and  the  radii 
of  the  circles  are  equal,  then  AB  is  greater  than  CD. 

Take  AME  equal  to  CND.  Join  AE,  OE,  and  OB. 
Then  AE  is  equal  to  CD  (185). 

Since  the  arc  AMB  is  less  than  a  semi-circumference, 
the  chord  AB  will  pass  between  the  arc  and  the  center 
0,     Hence,  it  cuts  the  radius  OE  at  some  point  I. 


62 


ELEMENTS    OF    GEOMETRY. 


Now,  the  broken  line   OIB  is  greater  than  OB  (54), 
or  its   equal  OE.      Subtracting   01  from   each   (8),  the 

C  N 


remainder  IB  is  greater  than  the  remainder  IE.  Add- 
ing AI  to  each  of  these,  we  have  AB  greater  than 
AIE.  But  AIE  is  greater  than  AE.  Therefore,  AB, 
the  chord  of  the  greater  arc,  is  greater  than  AE,  or  its 
equal  CD,  the  chord  of  the  less. 

lOO.  Corollary — When  the  arcs  are  both  greater  than 
a  semi-circumference,  the  greater  arc  has  the  less  chord. 


DISTANCE    FROM   THE   CENTER. 

lOl.  Theorem. —  When  the  radii  are  equal,  equal  chords 
are  equally  distant  from  the  center. 

Let  the  chords  AB  and  CD  be  equal,  and  in  the  equal 

0 


circles  ABG   and    CDF;    then   the    distances   of   these 
chords  from  the  centers  E  and  H  will  also  be  equal. 


CHORDS. 


63 


Let  fall  the  perpendiculars  EK  and  HL  from  the 
centers  upon  the  chords. 

Now,  since  the  chords  AB  and  CD  are  equal,  the  arcs 
AB  and  CD  are  also  equal  (185) ;  and  we  may  apply  the 
circle  ABG  to  its  equal  CDF,  so  that  they  will  coincide, 
and  the  arc  AB  coincide  with  its  equal  CD.  Therefore, 
the  chords  will  coincide.  Since  K  and  L  are  the  mid- 
dle points  of  these  coinciding  chords  (187),  K  will  fall 
upon  L.  Therefore,  the  lines  EK  and  HL  coincide  and 
are  equal.  But  these  equal  perpendiculars  measure  the 
distance  of  the  chords  from  the  centers  (105). 

If  the  equal  chords,  as  MO  and  AB,  are  in  the  same 
circle,  each  may  be  compared  with  the  equal  chord  CD 
of  the  equal  circle  CDF. 

Thus  it  may  be  proved  that  the  distances  NE  and  EK 
are  each  equal  to  HL,  and  therefore  equal  to  each  other. 

192.  Theorem. —  When  the  radii  are  equal,  the  less  of 
two  unequal  chords  is  the  farther  from  the  center. 

Let  AB  be   the   greater  of  two  chords,  and  FG  the 
less,  in  the  same  or   an 
equal  circle.    Then  FG  is 
farther  from   the   center 
than  AB. 

Take  the  arc  AC  equal 
to  FG.  Join  AC,  and 
from  the  center  D  let  fall 
the  perpendiculars  DE 
and  DN  upon  AB  and 
AC. 

Since    the   arc  AC   is 
less  than  AB,  the  chord  AB  will  be  between  AC  and 
the    center    D,   and   will    cut    the    perpendicular    DN. 
Then  DN,  the  whole,  is  greater  than  DH,  the  part  cut 
off;  and  DH  is  greater  than  DE  (104).     So  much  the 


(}4  ELEMENTS   OF   GEOMETRY. 

more   is  DN  greater  than  DE.     Therefore,  AC  and  its 
equal  FG  are  farther  from  the  center  than  AB. 

19S.  Corollary — Conversely  of  these  two  theorems, 
when  the  radii  are  equal,  chords  which  are  equally  dis- 
tant from  the  center  are  equal ;  and  of  two  chords  which 
are  unequally  distant  from  the  center,  the  one  nearer 
to  the  center  is  longer  than  the  other. 

194.  Problem  in  Drawing — To  Used  a  given  arc. 

Draw  the  chord  of  the  arc,  and  erect  a  perpendicular  at  its 
center. 

State  the  theorem  and  the  problems  in  drawing  here  used. 

195.  "The  most  simple  case  of  the  division  of  an  arc,  after 
its  bisection,  is  its  trisection,  or  its  d-ivision  into  three  equal  parts. 
Tills  problem  accordingly  exercised,  at  an  early  epoch  in  the  prog- 
ress of  geometrical  science,  the  ingenuity  of  mathematicians,  and 
has  become  memorable  in  the  history  of  geometrical  discovery, 
for  having  baffled  the  skill  of  the  most  illustrious  geometers. 

"Its  object  was  to  determine  means  of  dividing  any  given  arc 
into  three  equal  parts,  without  any  other  instruments  than  the 
rule  and  compasses  permitted  by  the  postulates  prefixed  to  Euclid's 
Elements.  Simple  as  the  problem  appears  to  be,  it  never  has  been 
solved,  and  probably  never  will  be,  under  the  above  conditions." 
— Lardners  Treatise. 


ANGLES    AT    THE    CENTER. 

l^G.  Angles  which  have  their  vertex  at  the  center 
of  a  circle  are  called,  for  this  reason,  angles  at  the  center. 
The  arc  between  the  sides  of  an  angle  is  called  the  in- 
tercepted arc  of  the  ajigle. 

197.  Theorem. —  The  radii  being  equal,  any  two  angles 
at  the  center  have  the  same  ratio  as  their  intercepted  arcs. 

This  theorem  presents  the  three  following  cases: 
1st.  If  the  arcs  are  equal,  the  angles  are  equal. 


ANGLES    AT    THE    CENTER. 


65 


For  the  arcs  may  be  placed  one  upon  the  other,  and 
will  coincide.  Then  BC  will  coincide  with  AO,  and  DC 
with  EG.     Thus  the  angles  may  coincide,  and  are  equal. 

The  converse  is  proved  in  the  same  manner. 


2d.  If  the  arcs  have  the  ratio  of  two  whole  numbers, 
the  angles  have  the  same  ratio. 

Suppose,  for  example,  the  arc  BD  :  arc  AE  : :  13  :  5. 


Then,  if  the  arc  BD  be  divided  into  thirteen  equal  parts, 
and  the  arc  AE  into  five  equal  parts,  these  small  arcs 
will  all  be  equal.  Let  radii  join  to  their  respective  cen- 
ters all  the  points  of  division. 

The  small  angles  at  the  center  thus  formed  are  all 
equal,  because  their  intercepted  arcs  are  equal.  But 
BCD  is  the  sum  of  thirteen,  and  AOE  of  five  of  these 
equal  angles.     Therefore, 

angle  BCD  :  angle  AOE  :  :  13  :  5 ; 

that  is,  the  angles  have  the  same  ratio  as  the  arcs. 

Geom. — 6 


66  ELEMENTS    OF    GEOMETRY. 

3d.  It  remains  to  be  proved,  that,  if  the  ratio  of  the 
arcs  can  not  be  expressed  by  two  whole  numbers,  the 
angles  have  still  the  same  ratio  as  the  arcs ;  or,  that 
the  radius  being  the  same,  the 

arc  BD  :  arc  AE  :  :  angle  BCD  :  angle  AOE. 

If  this  proportion  is  not  true,  then  the  first,  second, 

A 

/  ^\^  /        \        ^\ 


and  third  terms  being  unchanged,  the  fourth  term  is 
either  too  large  or  too  small.  We  will  prove  that  it  is 
neither.  If  it  were  too  large,  then  some  smaller  angle, 
as  AOI,  would  verify  the  proportion,  and 

arc  BD  :  arc  AE  :  :  angle  BCD  :  angle  AOI. 

Let  the  arc  BD  be  divided  into  equal  parts,  so  small 
that  each  of  them  shall  be  less  than  EI.  Let  one  of 
these  parts  be  applied  to  the  arc  AE,  beginning  at  A, 
and  marking  the  points  of  division.  One  of  those  points 
must  necessarily  fall  between  I  and  E,  say  at  the  point 
U.     Join  OU. 

Now,  by  this  construction,  the  arcs  BD  and  AU  have 
the  ratio  of  two  whole  numbers.     Therefore, 

arc  BD  :  arc  AU  :  :  angle  BCD  :  angle  AOU. 
These  last  two  proportions  may  be  written  thus  (19) ; 
arc  BD  :  angle  BCD  :  :  arc  AE  :  angle  AOI ; 
arc  BD  :  angle  BCD  :  :  arc  AU  :  angle  AOU. 


METHOD    OF    LIMITS.  61 

Therefore  (21), 

arc  AE  :  angle  AOI   :  :  arc  AU  :  angle  AOU; 
or  (19), 

arc  AE  :  arc  AU  :  :  angle  AOI  :  angle  AOU. 

But  this  last  proportion  is  impossible,  for  the  first 
antecedent  is  greater  than  its  consequent,  while  the 
second  antecedent  is  less  than  its  consequent.  There- 
fore, the  supposition  which  led  to  this  conclusion  is 
false,  and  the  fourth  term  of  the  proportion,  first  stated, 
is  not  too  large.  It  may  be  shown,  in  the  same  way, 
that  it  is  not  too  small. 

Therefore,  the  angle  AOE  is  the  true  fourth  term  of 
the  proportion,  and  it  is  proved  that  the  arc  BD  is  to 
the  arc  AE  as  the  angle  BCD  is  to  the  angle  AOE. 

DEMONSTRATION    BY    LIMITS. 

19S.  The  third  case  of  the  above  proposition  may  be 
demonstrated  in  a  different  manner,  which  requires  some 
explanation. 

We  have  this  definition  of  a  limit:  Let  a  magnitude 
vary  according  to  a  certain  law  which  causes  it  to  ap- 
proximate some  determinate  magnitude.  Suppose  the 
first  magnitude  can,  by  this  law,  approach  the  second 
indefinitely,  but  can  never  quite  reach  it.  Then  the 
second,  or  invariable  magnitude,  is  said  to  be  the  limit 
of  the  first,  or  variable  one. 

109.  Any  curve  may  be  treated  as  a  limit.  The 
straight  parts  of  a  broken  line,  having  all  its  vertices 
in  the  curve,  may  be  diminished  at  will,  and  the  broken 
line  made  to  approximate  the  curve  indefinitely.  Hence, 
a  curve  is  the  limit  of  those  broken  lines  which  have  all 
tlieir  vertices  in  the  curve. 


68 


ELEMENTS    OF  GEOMETRY. 


SOO.  The  arc  BC,  which  is  cut  off  bj  the  secant  AD, 
may  be  diminished  by  successive 
bisections,  keeping  the  remain- 
ders next  to  B.  Thus  AD,  re- 
volving on  the  point  B,  may 
approach  indefinitely  the  tan- 
gent EF.  Hence,  the  tangent 
at  any  point  of  a  curve  is  the 
limit  of  the  secants  which  may 
cut  the  curve  at  that  point. 

301.  The  principle  upon  which  all  reasoning  by  the 
method  of  limits  is  governed,  is  that,  whatever  is  true  up 
to  the  limit  is  true  at  the  limit.  We  admit  this  as  an 
axiom  of  reasoning,  because  we  can  not  conceive  it  to 
be  otherwise. 

Whatever  is  true  of  every  broken  line  having  its 
vertices  in  a  curve,  is  true  of  that  curve  also.  What- 
ever is  true  of  every  secant  passing  through  a  point 
of  a  curve,  is  true  of  the  tangent  at  that  point. 

We  do  not  say  that  the  arc  is  a  broken  line,  nor  that 
the  tangent  is  a  secant,  nor  that  an  arc  can  be  without 
extent;  but  that  the  curve  and  the  tangent  are  limits 
toward  which  variable  magnitudes  may  tend,  and  that 
whatever  is  true  all  the  way  to  within  the  least  pos- 
sible distance  of  a  certain  point,  is  true  at  that  point. 

202.  Having  proved  (first  and  second  parts,  197) 
that,  when  two  arcs  have  the  ratio 
of  two  whole  numbers,  the  angles 
at  the  center  have  the  same  ratio, 
we  may  then  suppose  that  the  ra- 
tio of  BD  to  BF  can  not  be  ex- 
pressed by  whole  numbers. 

Now,  if  we  divide  BF  into  two 
equal   parts,  the  point  of  division  will  be  at  a  certain 


METHOD    OF    INFINITES.  69 

distance  from  D.  We  may  conceive  the  arc  BF  to 
be  divided  into  any  number  of  equal  parts,  and  by  in- 
creasing this  number,  the  point  0,  the  point  of  division 
nearest  to  D,  may  be  made  to  approach  within  any  con- 
ceivable distance  of  D.  By  the  second  part  of  the 
theorem  (197),  it  is  proved  that 

arc  BO  :  arc  BF  :  :  angle  BCO  :  angle  BCF. 

Now,  although  the  arc  BD  is  itself  incommensurable 
with  BF,  yet  it  is  the  limit  of  the  arcs  BO,  and  the 
ansle  BCD  is  the  limit  of  the  angles  BCO.  Therefore, 
since  whatever  is  true  up  to  the  limit  is  true  at  the  limit, 

arc  BD  :  arc  BF  :  :  angle  BCD  :  angle  BCF. 

That  is,  the  intercepted  arcs  have  the  same  r?tio  as 
their  angles  at  the  center. 

METHOD    OF    INFINITES. 

203.  Modern  geometers  have  made  much  use  of  a 
kind  of  reasoning  which  may  be  called  the  method  of 
infinites.  It  consists  in  supposing  that  any  line  cf  def- 
inite extent  and  form  is  composed  of  an  infinite  num- 
ber of  infinitely  small  straight  lines. 

A  surface  is  supposed  to  consist  of  an  infinite  number 
of  infinitely  narrow  surfaces,  and  a  solid  of  an  infinite 
number  of  infinitely  thin  solids.  These  thin  solids,  nar- 
row surfaces,  and  small  lines,  are  called  infinitesimals. 

204.  The  reasoning  of  the  method  of  infinites  is 
substantially  the  same  in  its  logical  rigor  as  of  the 
method  of  limits.  The  method  of  infinites  is  a  p'uch 
abbreviated  form  of  the  method  of  limits. . 

The  student  must  be  careful  how  he  adopts  it.  For 
when  the  infinite  is  brought  into  an  argument  b}'  the 
unskillful,  the  conclusion  is  very  apt  to  be  absurd      It 


70  ELEMENTS    OF   GEOMETRY. 

is  sufficient  to  say,  that  where  the  method  of  limits  can 
be  used,  the  method  of  infinites  may  also  be  used  with- 
out error. 

The  method  of  infinites  has  also  been  called  the 
7nefhod  of  indivisibles.  Some  examples  of  its  use  will 
be  given  in  the  course  of  the  work. 

ARCS   AND    ANGLES. 

We  return  to  the  subject  of  angles  at  the  center. 
The  theorem  last  given  (197)  has  the  following 

!S05.  Corollary. — If  two  diameters  are  perpendicular 
to  each  other,  they  divide  the  whole  circumference  into 
four  equal  parts. 

206.  A  Quadrant  is  the  fourth 
part  of  a  circumference. 

20T.  Since  the  angle  at  the  cen- 
ter varies  as  the  intercepted  arc, 
mathematicians  have  adopted  the 
same  method  of  measuring  both  an- 
gles and  arcs.  As  a  right  angle  is 
the  unit  of  angles,  so  a  quadrant  of  a  certain  radius 
may  be  taken  as  the  standard  for  the  measurement  of 
arcs  that  have  the  same  radius. 

For  the  same  reason,  w^e  usually  say  that  the  inter- 
cepted arc  measures  the  angle  at  the  center.  Thus,  the 
right  angle  is  said  to  be  measured  by  the  quadrant; 
half  a  right  angle,  by  one-eighth  of  a'  circumference; 
and  so  on. 

'   '      APPLICATIONS. 

20S.  In  tlie  applications  of  geometry  to  practical  purposes, 
the  quadrant  and  tlie  riglit  angle  are  divided  into  ninety  equal 
parts,  each  of  which  is  called  a  degree.     Each  degree  is  marked 


ARCS    AND   ANGLES. 


71 


thus  °,  and  is  divided   into  sixty   minutes,   marked    thus  ^;    a-nd 
each  minute  is  divided  into  sixty  seconds,  marked  tjius  '\ 

Hence,  it  appears  that  there  are  in  an  entire  circumference,  or 
in  the  sum  of  all  the  successive  angles  about  a  point,  300°,  or 
21600^,  or  1296  000^^  Some  astronomers,  mostly  the  Frencli, 
divide  the  right  angle  and  tlie  quadrant  into  one  hundred  parts, 
each  of  these  into  one  hundred;   and  so  on. 

209,  Instruments  for  measuring  angles  are  founded  upon  the 
principle  that  arcs  are  proportional  to  angles.  Sucl)  instruments 
usually  consist  either  of  a  part  or  an  entire  circle  of  metal,  on 
the  surface  of  which  is  accurately  engraven  its  divisions  into  de- 
grees, etc.  Many  kinds  of  instruments  used  by  surveyors,  navi- 
gators, and  astronomers,  are  constructed  upon  this  principle. 

210.  An  instrument  called  a  protractor  is  used,  in  drawing, 
for  measuring  angles,  and  for  laying  down,  on  paper,  angles  of  any 
required  size.  It  consists  of  a  semicircle  of  brass  or  mica,  the 
circumference  of  which  is  divided  into  degrees  and  parts  of  a 
degree. 

PROBLEMS    IN    DRAWING. 


211.  Problem — To  draw  an  angle  equal  to  a  given  angle. 

Let  it  be  required  to  draw  a  line  making,  with  the  given  line 
BC,  an  angle  at  B  equal  to  the  given 
angle  A. 

With  A  as  a  center,  and  any  as- 
sumed radius  AD,  draw  the  arc  DE 
cutting  the  sides  of  the  angle  A. 
With  B  as  a  center,  and  the  same 
radius  as  before,  draw  an  arc  FG. 
Join  DE.  With  F  as  a  center,  and  a 
radius  equal  to  DE,  draw  an  arc  cut- 
ting FG  at  the  point  G.  Join  BG. 
Then  GBF  is  the  required  angle. 

For,  joining  FG,  the  arcs  DE  and  FG  have  equal  radii  and 
equal  chords,  and  therefore  are  equal  (185).  Hence,  they  sub- 
tend equal  angles  (197). 

212.  Corollary. — An  arc  equal  to  a  given  arc  may  be  drawn  in 
the  same  way. 


72  ELEMENTS    OF    GEOMETRY. 

!S13.  Problem — To  draw  an  angle  equal  to  the  sum  of 
two  given  angles. 

Let  A  and  B  be  the  given  an- 
gles. First,  make  the  angle  DCE 
equal  to  A,  and  then  at  C,  on  the 
line  CE,  draw  the  angle  ECF 
equal  to  B.  The  angle  FCD  is 
equal  to  ths  sum  of  A  and  B  (9). 

214.  Corollary. — In  a  similar  manner,  draw  an  angle  equal  to 
the  sum  of  several  given  angles;  also,  an  angle  equal  to  the  dif- 
ference of  two  given  angles ;  or,  an  angle  equal  to  the  supplement, 
or  to  the  complement  of  a  given  angle. 

215.  Corollary. — By  the  same  methods,  an  arc  may  be  drawn 
equal  to  the  difference  of  two  arcs  having  equal  radii,  or  equal  to 
the  sum  of  several  arcs. 

216.  Problem. — To  erect  a  perpendicular  to  a  given 
line  at  its  extreme  point,  without  producing  the  given  line. 

A  right  angle  may  be  made  separately,  and  then,  at  the  end  of 
the  given  line,  an  angle  be  made  equal  to  the  given  angle. 

This  is  the  method  universally  employed  by  mechanics  and 
draughtsmen  to  construe?  right  angles  and  perpendiculars  by  the 
use  of  the  square. 

SIT.  Problem — To  draw  a  line  through  a  given  point 
parallel  to  a  given  line. 

This  has  been  done  by  means  of  perpendiculars  (174).  Tt  may 
be  done  with  an  oblique  secant,  by  making  the  alternate  or  the 
corresponding  angles  equal, 

ARCS    INTERCEPTED    BY    PARALLELS. 

218.  An  arc  which  is  included  between  two  parallel 
lines,  or  between  the  sides  of  an  angle,  is  called  inter- 
cepted. 

210.  Theorem — Two  parallel  lines  intercept  equal  arcs 
of  a  circumference. 


INTERCEPTED    ARCS. 


75 


The  two  lines  may  be  both  secants,  or  both  tangents, 
or  one  a  secant  and  one  a  tangent. 

1st.  When   both  are  secants. 

The  arcs  AC  and  BD  inter- 
cepted by  the  parallels  AB  and 
CD  are  equal. 

For,  let  fall  from  the  center  0 
a  perpendicular  upon  CD,  and 
produce  it  to  the  circumference 
at  E.  Then  OE  is  also  perpendicular  to  AB  (127), 
Therefore,  the  arcs  EA  and  EB  are  equal  (187);  and  the 
arcs  EC  and  ED  are  equal.  Subtracting  the  first  from 
the  second,  there  remains  the  arc  AC  equal  to  the  arc  BD. 

2d.  When  one  is  a  tangent. 

Extend  the  radius  OE  to  the 
point  of  contact.  This  radius 
is  perpendicular  to  the  tangent 
AB  (183).  Hence,  it  is  perpen- 
dicular to  the  secant  CD  (127), 
and  therefore  it  bisects  the  arc 
CED  at  the  point  E  (187).  That 
is,  the  intercepted  arcs  EC  and  ED  are  equal. 

3d.  When  both  are  tangents. 

Extend  the  radii  OE  and  01  to  the  points  of  contact. 
These  radii  being   perpendicular      a  i?  b 

(183)  to  the  parallels,  must  (103 
and  127)  form  one  straight  line. 
Therefore,  EI  is  a  diameter,  and 
divides  (157)  the  circumference 
into  equal  parts.  But  these  equal 
parts  are  the  arcs  intercepted  by 
the  parallel  tangents.  C  I  D 

Therefore,  in  every  case,  the  arcs  intercepted  by  two 
parallels  are  equal. 
Oeom. — 7 


74 


ELEMENTS   OF   GEOMETRY. 


ARCS    INTERCEPTED    BY    ANGLES. 


320.  An  Inscribed  Angle  is  one  whose   sides  are 
chords  or  secants,  and  whose  vertex  is  on  the  circum- 
ference.    An  angle  is   said  to  be   inscribed  in  an  arc, 
when   its   vertex  is   on  the  arc 
and    its     sides     extend    to    or 
through    the   ends    of  the   arc. 
In  such  a  case  the  arc  is  said 
to  contain  the  angle.     Thus,  the 
angle  AEI  is   inscribed   in  the 
arc  AEI,  and  the  arc  AEI  con- 
tains the  angle  AEI. 

An  angle  is  said  to  stand  upon  the  arc  intercepted 
between  its  sides.  Thus,  the  angle  AEI  stands  upon  the 
arc  AOL 

221.  Corollary. — The  arc  in  which  an  angle  is  in- 
scribed, and  the  arc  intercepted  between  its  sides,  com- 
pose the  whole  circumference. 

222m  Theorem. — An  inscribed  angle  is  measured  by 
half  of  the  intercepted  arc. 

This  demonstration  also  presents  three  cases.  The 
center  of  the  circle  may  be  on  one  of  the  sides  of  the 
angle,  or  it  may  be  inside,  or  it  may  be  outside  of  the 
angle. 

1st.  One  side  of  the  angle,  as 
AB,  may  be  a  diameter. 

Make  the  diameter  DE,  paral- 
lel to  BC,  the  other  side  of  the 
angle.  Then  the  angle  B  is  equal 
to  its  alternate  angle  BOD  (125), 
which  is  measured  by  the  arc 
BD  (207).  This  arc  is  equal  to 
OE    (219),  and  also   to  EA  (197).     Therefore,  the  arc 


INTERCEPTED    ARCS. 


75 


BD  is  equal  to  the  half  of  AC,  and  the  inscribed  angle 
B  is  measured  by  half  of  its  intercepted  arc. 

2d.  The  center  of  the  circle  may  be  within  the  angle. 

From  the  vertex  B  extend  a  diameter  to  the  opposite 
side  of  the  circumference  at  D. 
As  just  proved,  the  angle  ABD 
is  measured  by  half  of  the  arc 
AD,  and  the  angle  DBC  by  half 
of  the  arc  DC.  Therefore,  the 
sum  of  the  two  angles,  or  ABC, 
is  measured  by  half  of  the  sum 
of  the  two  arcs,  or  half  of  the  arc  ADC. 

Bd.  The  center  of  the  circle  may  be  outside  of  the  angle. 

Extend  a  diameter  from  the 
vertex  as  before.  The  angle 
ABC  is  equal  to  ABD  diminished 
by  DBC,  and  is,  therefore,  meas- 
ured by  half  of  the  arc  DA  di- 
minished by  half  of  DC;  that  is, 
by  the  half  of  AC. 

23S.  Corollary. — When  an  inscribed  angle  and  an 
angle  at  the  center  have  the  same  intercepted  arc,  the 
inscribed  angle  is  half  of  the  angle  at  the  center. 


224.  Corollary. — All  angles  in- 
scribed in  the  same  arc  are  equal, 
for  they  have  the  same  measure. 


225.  Corollary. — Every  angle  inscribed  in  a  semi- 
circumference  is  a  right  angle.  If  the  arc  is  less  than 
a  semi-circumference,  the  angle  is  obtuse.  If  the  arc 
is  greater,  the  angle  is  acute. 


76 


ELEMENTS    OF    GEOMETRY 


!SS6.  Theorem — The  angle  formed  hij  a  tangent  and 
a  chord  is  measured  by  half  the  intercepted  arc. 

The  angle  CEI,  formed  by  the  tangent  AC  and  the 
chord  EI,  is  measured  by  half 
the  intercepted  arc  IDE. 

Through  I,  make  the  chord  10 
parallel  to  the  tangent  AC. 

The  angle  CEI  is  equal  to  its 
alternate  EIO  (125),  which  is 
measured  by  half  the  arc  OME 
(222),  which  is  equal  to  the  arc 
IDE  (219).  Therefore,  the  angle  CEI  is  measured  by 
half  the  arc  IDE. 

The  sum  of  the  angles  AEI  and  CEI  is  two  right 
angles,  and  is  therefore  measured  by  half  the  whole  cir- 
cumference (207).  Hence,  the  angle  AEI  is  equal  to 
two  right  angles  diminished  by  the  angle  CEI,  and  is 
measured  by  half  the  whole  circumference  diminished 
by  half  the  arc  IDE ;  that  is,  by  half  the  arc  lOME. 

Thus  it  is  proved  that  each  of  the  angles  formed  at 
E,  is  measured  by  half  the  arc  intercepted  between  its 
sides. 

227.  This  theorem  may  be  demonstrated  very  ele- 
gantly by  the  method  of  limits  (200). 

228.  Theorem. — Every  angle  whose  vertex  is  within 
the  circumference,  is  measured  hy 

half  the  sum  of  the  arcs  intercepted 

between  its  sides  and  its  sides  pro-       0/ 7-.---\U 

duced. 


Thus,  the  angle  OAE  is  meas- 
ured by  half  the  sum  of  the  arcs 
OE  and  lU. 

To  be  demonstrated  by  the 
student,  using  the  previous  theorems  (219  and  222). 


INTERCEPTED    ARCS. 


77 


S29.  Theorem — Every  angle  whose  vertex  is  outside 
of  a  circumference,  and  whose  sides  are  either  tangent  or 
secant,  is  measured  hy  half  the  difference  of  the  inter- 
cepted arcs. 

Thus,  the  angle  ACF  is  measured  by  half  the  dif- 
ference of  the  arcs  AF  and 
AB ;  the  angle  FCG,  by  half 
the  difference  of  the  arcs 
FG  and  BI;  and  the  angle 
ACE,  by  half  the  difference 
of  the  arcs  AFGE  and 
ABIE. 

This,  also,  may  be  demon- 
strated by  the  student,  by 
the  aid  of  the  previous  theo- 
rems on  intercepted  arcs. 


PROBLEMS    IN    DRAWING. 

S30.  Problem — Through  a  given  point  out  of  a  cir- 
cumference, to  draw  a  tangent  to  the  circumference. 

Let  A  be  the  given  point,  and  C  the  center  of  the  given  circle. 
Join  AC.  Bisect  AC  at  the 
point  B  (171).  With  B  as  a 
center  and  BC  as  a  radius, 
describe  a  circumference.  It 
will  pass  through  C  and  A 
(153),  and  will  cut  the  cir- 
cumference in  two  points,  D 
and  E.  Draw  straight  lines 
from  A  through  D  and  E. 
AD  and  AE  are  both  tangent 
to  the  given  circumference. 

Join  CD  and  CE.  The  angle  CDA  is  inscribed  in  a  semi- 
circumference,  and  is  therefore  a  right  angle  (225).  Since  AD  is 
perpendicular  to  the  radius  CD,  it  is  tangent  to  the  circumference 
(ITS).     AE  is  tangent  for  the  same  reasons. 


78 


ELEMENTS    OF    GEOMETRY. 


;S31.  Problem. —  Upon  a  given  chord  to  describe  an  arc 
which  shall  contain  a  given  angle. 

Let  AB  be  the  chord,  and  C  the  aM<;le.  Make  the  angle  DAB 
equal  to  C.  At  A  erect 
a  perpendicular  to  AD, 
and  erect  a  perpendicu- 
lar to  AB  at  its  center 
(172).  Produce  these  till 
they  meet  at  the  point 
F  (137).  With  F  as  a 
center,  and  FA  as  a  ra- 
dius, describe  a  circum- 
ference. Any  angle  in- 
scribed in  the  arc  BGHA 
will  be  equal  to  the  given 
angle  C. 

For  AD,  being  perpendicular  to  the  radius  FA,  is  a  tangent 
(178).  Therefore,  the  angle  BAD  is  measured  by  half  of  the  arc 
AIB  (226).  But  any  angle  contained  in  the  arc  AHGB  is  also 
measured  by  half  of  the  same  arc  (222),  and  is  therefore  equal 
to  BAD,  which  was  made  equal  to  C. 


POSITIONS    OF    TWO    CIRCUMFERENCES. 

232.  Theorem. — Two  circumferences  can  not  cut  each 
other  in  more  than  two  poiyits. 

For  three  points  determine  the  position  and  extent 
of  a  circumference  (149).  Therefore,  if  two  circumfer- 
ences have  three  points  common,  they  must  coincide 
throughout. 

S33.  Let  us  investigate  the  various  positions  which 
two  circumferences  may  have  with  reference  to  each 
other. 

Let  A  and  B  be  the  centers  of  two  circles,  and  let 
these  points  be  joined  by  a  straight  line,  w^hich  there- 
fore measures  the  distance  between  the  centers. 

First,  suppose  the  sum  of  the  radii  to  be  less  than  AB. 


TWO    CIRCUMFERENCES. 


79 


Then  AC  and  BD,  the  radii,  can  not  reach  each  other. 
At  C  and  D,  where  the  curves 
cut  tlie  line  AB,  let  perpendic- 
ulars to  that  line  be  erected. 
These  perpendiculars  are  paral- 
lel to  each  other  (129).  They 
are  also  tangent  respectively 
to  the  two  circumferences  (178).  It  follows,  therefore, 
that  CD,  the  distance  between  these  parallels,  is  also 
the  least  distance  between  the  two  curves. 

S^.  Next,  let  the  sum  of  the  radii  AC  and  BC  be 
equal  to  AB,  the  distance 
between  the  centers.  Then 
both  curves  will  pass  through 
the  point  C  (153).  At  this 
point  let  a  perpendicular  be 
erected  as  before.  This  per- 
pendicular CG  is  tangent  to 
both  the  curves  (178);  that  is,  it  is  cut  by  neither  of 
them.  Therefore,  the  curves  have  only  one  common 
point  C. 

235.  Next,  let  AB  be  less  than  the  sum,  but  greater 
than  the  diiFerence,  of  the  radii 
AC  and  BD.  Then  the  point 
C  will  fall  within  the  circum- 
ference DF.  For  if  it  fell 
on  or  outside  of  it,  on  the 
side  toward  A,  then  AB  would 
be  equal  to  or  greater  than 
the  sum  of  the  radii ;  and  if  the  point  C  fell  on  or  out- 
side of  the  curve  in  the  direction  toward  B,  then  AB 
would  be  equal  to  or  less  than  the  difference  between 
the  radii.  Each  of  these  is  contrary  to  the  hypothesis. 
For  the  same  reasons,  the  point  D  will  fall  within  the 


80  ELEMENTS   OF   GEOMETRY 

circumference  CG.     Therefore,  these  circumferences  cut 
each  other,  and  have  two  points  common  (232). 

S36.  Next,  let  the  difference  between  the  two  radii 
AC  and  BC  be  equal  to  the 
distance  AB.  A  perpendic- 
ular to  this  line  at  the  point 
C  will  be  a  tangent  to  both 
curves,  and  they  have  a  com- 
mon point  at  C.  They  have 
no  other  common  point,  for 
the  two  curves  are  both  symmetrical  about  the  line  AC 
(158),  and,  therefore,  if  they  had  a  common  point  on  one 
side  of  that  line,  they  would  have  a  corresponding  com- 
mon point  on  the  other  side;  but  this  can  not  be,  for 
they  would  then  have  three  points  common  (232). 

2S7*  Lastly,  suppose  the  distance  AB  less  than  the 
difference  of  the  radii  AC 
and  BD,  by  the  line  CD. 
That  is, 

AB  +  BD  +  DC  =  AC.      A— ^ 

Join  A,  the  center  of  the 
larger  circle,  with  F,  any 
point  of  the  smaller  cir- 
cumference, and  join  BF.  Then  AB  and  BD  are  to- 
gether equal  to  AB  and  BF,  which  are  together  greater 
than  AF.  Therefore,  AD  is  greater  than  AF.  Hence, 
the  point  D  is  farther  from  A  than  any  other  point  of 
the  circumference  DF.  It  follows  that  CD  is  the  least 
distance  between  the  two  curves. 

The  above  course  of  reasoning  develops  the  follow- 
ing principles : 

)S38.  Theorem.  —  Two  circumferences  may  have,  with 
reference  to  each  other,  five  positions: 


TWO    CIRCUMFERENCES.  81 

1st.  Each  may  he  entirely  exterior  to  the  other,  when  the 
distance  between  their  centers  is  greater  than  the  su:n  of 
their  radii. 

2d.  They  may  touch  each  other  exteriorly,  having  one 
point  common,  when  the  distance  between  the  centers  is 
equal  to  the  sum  of  the  radii. 

Sd.  They  may  cut  each  other,  having  two  points  com- 
mon, when  the  distance  between  the  centers  is  less  than  the 
sum  and  greater  than  the  difference  of  the  radii. 

4th.  One  may  be  within  the  other  and  tangent,  having 
one  point  common,  when  the  distance  between  the  centers  is 
equal  to  the  difference  of  the  radii. 

bth.  One  may  be  entirely  within  the  other,  tvhen  the 
distance  between  the  centers  is  less  than  the  difference  of 
the  radii. 

S30.  Corollary — Two  circumferences  can  not  have 
more  than  one  chord  common  to  both. 

:^0.  Corollary — The  common  chord  of  two  circum- 
ferences is  perpendicular  to  the  straight  line  which  joins 
their  centers  and  is  bisected  by  it.  For  the  ends  of  the 
chords  are  equidistant  from  each  of  the  centers,  the 
ends  of  the  other  line  (109). 

S4].  Corollary. — When  two  circumferences  are  tan- 
gent to  each  other,  the  two  centers  and  the  point  of 
contact  are  in  one  straight  line. 

;^12.  Corollary. — When  two  circumferences  have  no 
common  point,  the  least  distance  between  the  curves  is 
measured  along  the  line  which  joins  the  centers. 

tMS.  Corollary. — When  the  distance  between  the  cen- 
ters is  zero,  that  is,  when  they  coincide,  a  straight  line 
through  this  point  may  have  any  direction  in  the  plane; 
and  the  two  curves  are  equidistant  at  all  points.  Such 
circles  are  called  Concentric. 


82  ELEMENTS   O;-    GEOMETRY. 

214.  A  Locus  is  a  line  or  a  surfivce  all  the  points 
of  which  have  some  common  property,  which  does  not 
belong  to  any  other  points.  This  is  also  frequently 
called  a  geometrical  locus.     Thus, 

The  circumference  of  a  circle  is  the  locus  of  all  those 
points  in  the  plane,  which  are  at  a  given  distance  from 
a  given  point. 

A  straight  line  perpendicular  to  another  at  its  center 
is  the  locus  of  all  those  points  in  the  plane,  which  are  at 
the  same  distance  from  both  ends  of  the  second  line. 

The  geometrical  locus  of  the  centers  of  those  circles 
which  have  a  given  radius,  and  are  tangent  to  a  given 
straight  line,  is  a  line  parallel  to  the  former,  and  at  a 
distance  from  it  equal  to  the  radius. 

24:5«  The  student  will  find  an  excellent  review  of  the 
preceding  pages,  in  demonstrating  the  theorems,  and 
solving  the  problems  in  drawing  which  follow. 

In  his  efforts  to  discover  the  solutions  of  the  more 
difficult  problems  in  drawing,  the  student  will  be  much 
assisted  by  the  following 

Suggestions. — 1.  Suppose  the  problem  solved,  and  the 
figure  completed. 

2.  Find  the  geometrical  relations  of  the  different 
parts  of  the  figure  thus  formed,  drawing  auxiliary  lines, 
if  necessary. 

3.  From  the  principles  thus  developed,  make  a  rule 
for  the  solution  of  a  problem. 

This  is  the  analytic  method  of  solving  problems. 

EXERCISES. 

1.  Take  two  straight  lines  at  random,  and  find  their  ratio. 
Make  examples  in  this  way  for  all  the  problems  in  drawing. 

2.  Bisect  a  quadrant,  also  its  half,  its  fourth;  and  so  on. 


EXERCISES.  83 

3.  From  a  given  point,  to  draw  the  shortest  line  possible  to  a 
given  straight  line. 

4.  With  a  given  length  of  radius,  to  draw  a  circumference 
through  two  given  points. 

5.  From  two  given  points,  to  draw  two  equal  straight  lines 
which  shall  end  in  the  same  point  of  a  given  line. 

6.  From  a  point  out  of  a  straight  line,  to  draw  a  second  lii  e 
making  a  required  angle  with  the  first. 

7.  If  from  a  point  without  a  circle  two  straight  lines  extend  to 
the  concave  part  of  the  circumference,  making  equal  angles  with 
the  line  joining  the  same  point  and  the  center  of  the  circle,  then 
the  parts  of  the  first  two  lines  which  are  within  the  circumfer- 
ence are  equal. 

8.  To  draw  a  line  through  a  point  such  that  the  perpendicu- 
lars upon  this  line,  from  two  other  points,  may  be  equal. 

9.  From  two  points  on  the  same  side  of  a  straight  line,  to 
draw  two  other  straight  lines  which  shall  meet  in  the  first,  and 
make  equal  angles  with  it. 

10.  In  each  of  the  five  cases  of  the  last  theorem  (238),  how 
many  straight  lines  can  be  tangent  to  both  circumferences? 

The  number  is  different  for  each  case. 

11.  On  any  two  circumferences,  the  two  points  which  are  at  the 
greatest  distance  apart  are  in  the  prolongation  of  the  line  which 
joins  the  centers. 

12.  To  draw  a  circumference  with  a  given  radius,  through  a 
given  point,  and  tangent  to  a  given  straight  line. 

13.  With  a  given  radius,  to  draw  a  circumference  tangent  to 
two  given  circumferences. 

14.  What  is  the  locus  of  the  centers  of  those  circles  which  have 
a  given  radius,  and  are  tangent  to  a  given  circle? 

15.  Of  all  straight  lines  which  can  be  drawn  from  two  given 
points  to  meet  on  the  convex  circumference  of  a  circle,  the  sum 
of  those  two  is  the  least  which  make  equal  angles  with  the  tan- 
gent to  the  circle  at  the  point  of  concourse. 

16.  If  two  circumferences  be  such  that  the  radius  of  one  is 
the  diameter  of  the  other,  any  straight  line  extending  from  their 
point  of  contact  to  the  outer  circumference  is  bisected  by  the 
inner  one. 


84  ELEMENTS    OF  GEOMETRY. 

17.  If  two  circumferences  cut  each  other,  and  from  either  point 
of  intersection  a  diameter  be  made  in  each,  the  extremities  of 
these  diameters  and  the  other  point  of  intersection  are  in  the  same 
straight  line. 

18.  If  any  straight  line  joining  two  parallel  lines  be  bisected, 
any  other  line  through  the  point  of  bisection  and  joining  the  two 
parallels,  is  also  bisected  at  that  point. 

19.  If  two  circumferences  are  concentric,  a  line  which  is  a 
chord  of  the  one  and  a  tangent  of  the  other,  is  bisected  at  the 
point  of  contact. 

20.  If  a  circle  have  any  number  of  equal  chords,  what  is  the 
locus  of  their  points  of  bisection? 

21.  If  any  point,  not  the  center,  be  taken  in  a  diameter  of  a 
circle,  of  all  the  chords  which  can  pass  through  that  point,  that 
one  is  the  least  which  is  at  right  angles  to  the  diameter. 

22.  If  from  any  point  there  extend  two  lines  tangent  to  a 
circumference,  the  angle  contained  by  the  tangents  is  double  the 
angle  contained  by  the  line  joining  the  points  of  contact  and  the 
radius  extending  to  one  of  them. 

23.  If  from  the  ends  of  a  diameter  perpendiculars  be  let  fall 
on  any  line  cutting  the  circumference,  the  parts  intercepted  be- 
tween those  perpendiculars  and  the  curve  are  equal. 

24.  To  draw  a  circumference  with  a  given  radius,  so  (hat  the 
eiaes  of  a  given  angle  shall  be  tangents  to  it. 

25.  To  draw  a  circumference  through  two  given  poi.ivs.,  with  the 
center  in  a  given  line. 

26.  Through  a  given  point,  to  draw  a  straig\iv  line,  making 
equal  angles  with  the  two  sides  of  a  given  angle. 


PROPERTIES   OF   TRIANGLES.  85 


CHAPTER    V. 

TRIANGLES. 

246.  Next  in  regular  order  is  the  consideration  of 
those  plane  figures  which  inclose  an  area ;  and,  first,  of 
those  whose  boundaries  are  straight  lines. 

A  Polygon  is  a  portion  of  a  plane  bounded  by 
straight  lines.  The  straight  lines  are  the  sides  of  the 
polygon. 

The  Perimeter  of  a  polygon  is  its  boundary,  or  the 
sum  of  all  the  sides.  Sometimes  this  word  is  used  to 
designate  the  boundary  of  any  plane  figure. 

S47.  A  Triangle  is  a  polygon  of  three  sides. 

Less  than  three  straight  lines  can  not  inclose  a  sur- 
face, for  two  straight  lines  can  have  only  one  common 
point  (51).  Therefore,  the  triangle  is  the  simplest 
polygon.  From  a  consideration  of  its  properties,  those 
of  all  other  polygons  may  be  derived. 

248.  Problem. — Any  three  points  not  in  the  same 
straight  line  may  he  made  the  vertices  of  the  three  angles 
of  a  triangle. 

For  these  points  determine  the  plane  (60),  and  straight 
lines  may  join  them  two  and  two  (47),  thus  forming  the 
required  figure. 

INSCBIBED    AND   CIRCUMSCRIBED. 

249.  Corollary — Any  three  points  of  a  circumference 
may  be  made  the  vertices  of  a  triangle.     A  circumfer- 


m  ELEMENTS   OF   GEOxMETRY. 

ence  may  pass  through  the  vertices  of  any  triangle,  for 
it  may  pass  through  any  three  points  not  in  the  same 
straight  line  (149). 

S50.  Theorem. —  Within  every  triangle  there  is  a  point 
equally  distant  from  the  three  sides. 

In  the  triangle  ABC,  let  lines  bisecting  the  angles  A 
and  B  be  produced  until  they 
meet. 

The  point  D,  where  the  two 
bisecting  lines  meet,  is  equally 
distant  from  the  two  sides  AB 
and  BC,  since  it  is  a  point  of 

the  line  which  bisects  the  angle  B  (113).  For  a  similar 
reason,  the  point  D  is  equally  distant  from  the  two 
sides  AB  and  AC.  Therefore,  it  is  equally  distant 
from  the  three  sides  of  the  triangle. 

251.  Corollary. — The  three  lines  which  bisect  the  sev- 
eral angles  of  a  triangle  meet  at  one  point.  For  the 
point   D   must  be  in   the   line  which  bisects  the   angle 

C  (113). 

252.  Corollary — With  D  as  a  center,  and  a  radius 
equal  to  the  distance  of  D  from  either  side,  a  circum- 
ference may  be  described,  to  which  every  side  of  the 
triangle  will  be  a  tangent. 

25S.  When  a  circumference  passes  through  the  ver- 
tices of  all  the  angles  of  a  polygon,  the  circle  is  said  to 
be  circumscribed  about  the  polygon,  and  the  polygon  to 
be  inscribed  in  the  circle.  When  every  side  of  a  polygon 
is  tangent  to  a  circumference,  the  circle  is  inscribed  and 
the  polygon  circumscribed. 

254.  The  angles  at  the  ends  of  J? 

one  side  of  a  triangle  are  said  to 
be  adjacent  to  that  side.     Thus,  the 


PROPERTIES    OF   TRIANGLES.  8^ 

an^-lc3  A  and  B  are  adjacent  to  the  side  AB.  The 
angle  formed  by  the  other  two  sides  is  opposite.  Thus, 
the  angle  A  and  the  side  BC  are  opposite  to  each 
other. 


SUM    OF    THE    ANGLES. 

255.  Theorem — The  sum  of  the  angles  of  a  triangle 
is  equal  to  two  right  angles. 

Let  the  line  DE  pass  through  the  vertex  of  one  an- 
gle, B,  parallel  to  the  op-         ^                   BE 
posite  side,  AC.  ■••p^ 

Then  the  angle  A  is  equal  ^/^        \ 

to  its  alternate  angle  DBA        A'^— ^C 

(125).  For  the  same  rea- 
son, the  angle  C  is  equal  to  the  angle  EBC.  Hence, 
the  three  angles  of  the  triangle  are  equal  to  the  three 
consecutive  angles  at  the  point  B,  which  are  equal  to 
two  right  angles  (91).  Therefore,  the  sum  of  the  three 
angles  of  the  triangle  is  equal  to  two  right  angles. 

!^6.  Corollary. — Each  angle  of  a  triangle  is  the  sup- 
plement of  the  sum  of  the  other  two. 

257.  Corollary. — At  least  two  of  the  angles  of  a  tri- 
angle are  acute. 

258.  Corollary. — If  two  angles  of  a  triangle  are  equal, 
they  are  both  acute.  If  the  three  are  equal,  they  are 
all  acute,  and  each  is  two-thirds  of  a  right  angle. 

259.  An  Acute  Angled  triangle 
is  one  which  has  all  its  angles  acute, 
as  a. 

A  Right  Axgled  triangle  has  one 
of  the  ancrles  riorht,  as  h. 


88  ELEMENTS    OF   GEOMETRY. 

An  Obtuse  Angled   triangle   has 
one  of  the  angles  obtuse,  as  c. 

260.  Corollary — In  a  right  angled  triangle,  the  two 
acute  angles  are  complementary  (94). 

261.  Corollary — If  one   side   of  a  triangle  be   pro- 
duced, the  exterior  angle  thus  B 
formed,   as   BCD,  is   equal   to 
the    sum   of   the   two   interior 
angles  not  adjacent  to  it,  as  A     A- 
and  B  (256).     So  much  the  more,  the  exterior  angle  is 
greater  than  either  one  of  the  interior  angles  not  adja- 
cent to  it. 

302.  Corollary. — If  two  angles  of  a  triangle  are  re- 
spectively equal  to  two  angles  of  another,  then  the  third 
angles  are  also  equal. 

26^.  Either  side  of  a  triangle  may  be  taken  as  the 
hase.  Then  the  vertex  of  the  angle  opposite* the  base 
is  the  vertex  of  the  triangle. 

The  Altitude  of  the  triangle  is  the  distance  from 
the  vertex  to  the  base,  which  is  measured  by  a  perpen- 
dicular let  fall  on  the  base  produced,  if  necessary. 

264.  Corollary — The  altitude  of  a  triangle  is  equal 
to  the  distance  between  the  base  and  a  line  through  the 
vertex  parallel  to  the  base. 

265.  .When  one  of  the  angles  at  the  base  is  obtuse, 
the  perpendicular  falls  outside  of  the  triangle. 

When  one  of  the  angles  at  the  base  is  right,  the  alti- 
tude coincides  with  the  perpendicular  side. 

When  both  the  angles  at  the  base  are  acute,  the  alti- 
tude falls  within  the  triangle. 

Let  the  student  give  the  reason  for  each  case,  and 
illustrate  it  with  a  diagram. 


PROPERTIES    OF   TRIANGLES.  &9 

LIMITS    OF     SIDES. 

S66.  Theorem. — Each  side  of  a  triangle  is  smaller 
than  the  sum  of  the  other  two,  and  greater  than  their  dif- 
ference. 

The  first  part  of  this  theorem  is  an  immediate  conse- 
quence of  the  Axiom  of  Distance 
(54) ;  that  is, 

AC  <  AB  +  BC.  ^ 

Subtract  AB  from  both  members  of  this  inequality,  and 

AC  —  AB  <  BC. 

That  is,  BC  is  greater  than  the  diiference  of  the  other 
sides. 

Prove  the  same  for  each  of  the  other  sides. 

267.  An  Equilateral  triangle  is  one  which  has 
three  sides  equal. 

An  Isosceles  triangle  is  one  which  has  only  two  sides 
equal. 

A  Scalene  triangle  is  one  which  has  no  two  sides 
equal. 

EQUAL    SIDES. 

268.  Theorem. —  When  two  sides  of  a  triangle  are 
equal,  the  angles  opposite  to  them  are  equal. 

If  the  triangle  BCD  is  isosceles,  the  angles  B  and  D, 
which  are  opposite  the  equal  sides, 
are  equal. 

Let  the  angle  C  be  divided  into 
two  equal  parts,  and  let  the  divid- 
ing line  extend  to  the  opposite  side 
of  the  triangle  at  F. 

Then,  that  portion  of  the  figure 
upon  one  side  of  this  line  may  be  turned  upon  it  as 
Geom.— 8 


90  ELEMENTS    OF    GEOMETRY. 

upon  an  axis.  Since  the  angle  C  was  bisected,  the  line 
BC  will  fall  upon  DC;  and,  since  these  two  lines  are 
equal,  the  point  B  will  fall  upon  D.  But  F,  being  a 
point  of  the  axis,  remains  fixed;  hence,  BF  and  DF 
will  coincide.  Therefore,  the  angles  B  and  D  coincide, 
and  are  equal. 

2GO.  Corollary — The  three  angles  of  an  equilateral 
triangle  are  equal. 

270.  In  an  isosceles  triangle,  the  angle  included  by 
the  equal  sides  is  usually  called  the  vertex  of  the  trian- 
gle, and  the  side  opposite  to  it  the  base. 

ISTl.  Corollary.— If  a  line  pass  through  the  vertex  of 
an  isosceles  triangle,  and  also  through  the  middle  of 
the  base,  it  will  bisect  the  angle  at  the  vertex,  and  be 
perpendicular  to  the  base. 

The  straight  line  which  has  any  two  of  these  four  con- 
ditions must  have  the  other  two  (52). 

UNEQUAL    SIDES. 

2T2.  Theorem. —  When  hvo  sides  of  a  triangle  are  une- 
qual^ the  angle  ojjposife  to  the  greater  side  is  greater  than 
the  angle  opposite  to  the  less  side. 

If  in  the  triangle  BCD  the  side  BC  is  greater  than 
DC,  then  the  angle  D  is  greater 
than  the  angle  B. 

Let  the  line  CF  bisect  the  an- 
gle C,  and  be  produced  to  the  side 
BD.     Then  let  the  triangle  CDF 
turn  upon  CF.     CD  will  take  the     g^ 
direction  CB  ;  but,  since  CD  is  less 

than  CB,  the  point  D  will  fall  between  C  and  B,  at  G. 
Join  GF. 

Now,  the  angle  FGC  is  equal  to  the  angle  D,  because 


PROPERTIES    OF    TRIANGLES  91 

they  coincide;  and  it  is  greater  than  the  angle  B,  be- 
cause it  is  exterior  to  the  triangle  BGF  (261).  There- 
fore, the  angle  D  is  greater  than  B. 

273.  Corollary — When  one  side  of  a  triangle  is  not 
the  largest,  the  angle  which  is  opposite  to  that  side  is 
acute  (257). 

274.  Corollary. — In  a  scalene  triangle,  no  two  angles 
are  equal. 

EQUAL    ANGLES. 

275.  Theorem — If  two  angles  of  a  triangle  are  equal, 
the  sides  opposite  them  are  equal. 

For  if  these  sides  were  unequal,  the  angles  opposite 
to  them  would  be  unequal  (272),  which  is  contrary  to 
the  hypothesis. 

276.  Corollary — If  a  triangle  is  equiangular,  that  is, 
has  all  its  angles  equal,  then  it  is  equilateral. 

UNEQUAL    ANGLES. 

277.  Theorem — If  two  angles  of  a  triangle  are  une- 
qual, the  side  opposite  to  the  greater  angle  is  greater  than 
the  side  opposite  to  the  less. 

If,  in  the  triangle  ABC,  the  angle  C  is  greater  than 
the    angle    A,    then    AB    is  g 

greater  than  BC. 

For,  if  AB  were  not  greater 
than  BC,  it  would  be  either 
equal  to  it  or  less.  If  AB  were  equal  to  BC,  the  oppo- 
site angles  A  and  C  would  be  equal  (268) ;  and  if  AB 
Avere  less  than  BC,  then  the  angle  C  would  be  less  than 
A  (272) ;  but  both  of  these  co.nclusions  are  contrary  to 
the  hypothesis.  Therefore,  AB  being  neither  less  than 
nor  equal  to  BC,  must  be  greater. 


92  ELEMENTS   OF   GEOxVIETRY. 

2T8.  Corollary. — In  an  obtuse  angled  triangle,  the 
longest  side  is  opposite  the  obtuse  angle;  and  in  a  right 
angled  triangle,  the  longest  side  is  opposite  the  right 
angle. 

279.  The  Hypotenuse  of  a  right  angled  triangle  is 
the  side  opposite  the  right  angle.  The  other  two  sides 
are  called  the  legs. 

The  student  will  notice  that  some  of  the  above  prop- 
ositions are  but  different  statements  of  the  principles 
of  perpendicular  and  oblique  lines. 

EXERCISES. 

280. — 1.  How  many  degrees  are  there  in  an  angle  of  an  equi- 
lateral triangle? 

2.  If  one  of  the  angles  at  the  base  of  an  isosceles  triangle  be 
double  the  angle  at  the  vertex,  how  many  degrees  in  each  ? 

3.  If  the  angle  at  the  vertex  of  an  isosceles  triangle  be  double 
one  of  the  angles  at  the  base,  what  is  the  angle  at  the  vertex? 

4.  To  circumscribe  a  circle  about  a  given  triangle  (149). 

5.  To  inscribe  a  circle  in  a  given  triangle  (252). 

6.  If  two  sides  of  a  triangle  be  produced,  the  lines  which  bi- 
sect the  two  exterior  angles  and  the  third  interior  angle  all  meet 
in  one  point. 

7.  Draw  a  line  BE  parallel  to  the  base  BC  of  a  triangle  ABC, 
80  that  DE  shall  be  equal  to  the  sum  of  BD  and  CE. 

8.  Can  a  triangular  field  have  one  side  436  yards,  the  second 
547  yards,  and  the  third  984  yards  long? 

9.  The  angle  at  the  base  of  an  isosceles  triaigle  being  one- 
fourth  of  the  angle  at  the  vertex,  if  a  perpendicular  be  erected  to 
the  base  at  its  extreme  point,  and  this  perpendicular  meet  the 
opposite  side  of  the  triangle  produced,  then  the  part  produced,  the 
remaining  side,  and  the  perpendicular  form  an  equilateral  triangle. 

.10.  If  with  the  vertex  of  an  isosceles  triangle  as  a  center,  a 
circumference  be  drawn  cutting  the  base  or  the  base  produced, 
then  the  parts  intercepted  between  the  curve  and  the  extreniities 
of  the  base,  are  equal. 


EQUALITY    OF    TRIANGLES. 


93 


EQUALITY    OF    TRIANGLES. 

S81.  The  three  sides  and  three  angles  of  a  trian- 
gle may  be  called  its  six  elements.  It  may  be  shown 
that  three  of  these  are  always  necessary,  and  they  are 
generally  enough,  to  determine  the  triangle. 


THREE    SIDES    EQUAL. 

282,  Theorem. — Two  triangles  are  equal  when  the  three 
sides  of  the  one  are  respectively  equal  to  the  three  sides  of 
the  other. 

Let  the  side  BD  be  equal  to  AI,  the  side  BC  equal  to 
AE,  and  CD  to  EI ;  then  the 
two  triangles  are  equal. 

Apply  the  line  AI  to  its 
equal  BD,  so  that  the  point 
A  will  fall  upon  B.  Then 
I  will  fall  upon  D,  since  the 
lines  are  equal.  Next,  turn 
one  of  the  triangles,  if  nec- 
essary, so  that  both  shall 
fall  on  the  same  side  of  this 
common  line. 

Now,  the  point  A  being 
on  B,  the  points  E  and  C  are  at  the  same  distance  from 
B,  and  therefore  they  are  both  in  the  circumference, 
which  has  B  for  its  center,  and  BC  or  AE  for  its  ra- 
dius (153).  For  a  similar  reason,  the  points  E  and  C 
are  both  in  the  circumference,  w^hich  has  D  for  its  cen- 
ter and  DC  or  IE  for  its  radius.  These  two  circumfer- 
ences have  only  one  point  common  on  one  side  of  the 
line  BD,  which  joins  their  centers  (232).  Hence.  E  and 
C  are  both  at  this  point.     Therefore  (51),  AE  coincides 


94  ELE^iENTS   OF   GEOMETRY. 

with  BC,  and  EI  with  CD;    that  is,  the  two  triangles 
coincide  throughout,  and  are  equal. 

283.  Every  plane  figure  may  be  supposed  to  have 
two  faces,  which  may  be  termed  the  upward  and  the 
downward  faces.  In  order  to  place  the  triangle  m  upon 
Z,  we  may  conceive  it  to  slide  along  the  plane  without 
turning  over;  but,  in  order  to  place  n  upon  Z,  it  must  be 
turned  over,  so  that  its  upward  face  will  be  upon  the 
upward  face  of  I. 

There  are,  then,  two  methods  of  superposition;  the 
first,  called  direct,  when  the  downward  face  of  one  figure 
is  applied  to  the  upward  face  of  the  other;  and  the 
second,  called  inverse,  when  the  upward  faces  of  the  two 
are  applied  to  each  other.  Hitherto,  we  have  used 
only  the  inverse  method.  Generally,  in  the  chapter  on 
the  circumference,  either  method  might  be  used  indif- 
ferently. 

TWO    SIDES    AND    INCLUDED   ANGLE. 

284.  Theorem — Two  triangles  are  equal  when  they 
have  two  sides  and  the  included  angle  of  the  one,  respect- 
ively equal  to  two  sides  and  the  included  angle  of  the 
other. 

If  the  angle  A  is   equal  to  D,  and  the  side  AB  to 


the  side  DF,  and  AC  to  DE,  then  the  two  triangles  are 
equal. 

Apply  the  side  AC  to  its  equal  DE,  turning  one  tri- 


EQUALITY    OF    TRIANGLES.  95 

angle,  if  necessary,  so  that  both  shall  fall  upon  the  same 
side  of  that  common  line. 

Then,  since  the  angles  A  and  D  are  equal,  AB  must 
take  the  direction  DF,  and  these  lines  being  equal,  B 
will  fall  upon  F.  Therefore,  BC  and  FE,  having  two 
points  common,  coincide;  and  the  two  triangles  coincide 
throughout,  and  are  equal. 

ONE    SIDE    AND    TWO    ANGLES. 

2H5m  Theorem. — Two  triangles  are  equal  when  they 
have  one  side  and  two  adjacent  angles  of  the  one,  respect- 
ively equal  to  a  side  and  the  two  adjacent  angles  of  the 
other. 

If  the  triangles  ABC   and  DEF  have  the  side  AC 


equal  to  DE,  and  the  angle  A  equal  to  D,  and  C  equal 
to  E,  then  the  triangles  are  equal. 

Apply  the  side  AC  to  its  equal  DE,  so  that  the  ver- 
tices of  the  equal  angles  shall  come  together,  A  upon 
D,  and  C  upon  E,  and  turning  one  triangle,  if  neces- 
sary, so  that  both  shall  fall  upon  one  side  of  the  com- 
mon line. 

Then,  since  the  angles  A  and  D  are  equal,  AB  will 
take  the  direction  DF,  and  the  point  B  will  fall  some- 
where in  the  line  DF.  Since  the  angles  C  and  E  are 
equal,  CB  will  take  the  direction  EF,  and  B  will  also 
be  in  the  line  EF.  Therefore,  B  ftvlls  upon  F,  the  only 
point  common  to  the  two  lines  DF  and  EF.     Hence,  the 


96  ELEMENTS   OF    GEOMETRY. 

sides    of  the   one   triangle   coincide   with   those   of  the 
other,  and  the  two  triangles  are  equal. 

S88.  Theorem — Two  triangles  are  equal  when  they 
have  one  side  and  any  two  angles  of  the  one,  respectively 
equal  to  the  corresponding  parts  of  the  other. 

For  the  third  angle  of  the  first  triangle  must  be  equal 
to  the  third  angle  of  the  other  (262).  Then  this  be- 
comes a  case  of  the  preceding  theorem. 

TWO   SIDES   AND    AN    OPPOSITE   ANGLE. 

287.  Theorem — Two  triangles  are  equal  when  one  of 
them  has  two  sides,  and  the  angle  opposite  to  the  side 
which  is  equal  to  or  greater  than  the  other,  respectively 
equal  to  the  corresponding  parts  of  the  other  triangle. 

Let   the   sides   AE   and    EI,  EI  being   equal   to   or 

C 


greater  than  AE,  and  the  angle  A,  be  respectively  equal 
to  the  sides  BC,  CD,  and  the  angle  B.  Then  the  tri- 
angles are  equal. 

For  the  side  AE  may  be  placed  on  its  equal  BC. 
Then,  since  the  angles  A  and  B  are  equal,  AI  will  take 
the  direction  BD,  and  the  points  I  and  D  will  both  be 
in  the  common  line  BD.  Since  EI  and  CD  are  equal, 
the  points  I  and  D  are  both  in  the  circumference  whose 
center  is  at  C,  and  whose  radius  is  equal  to  CD.  Now, 
this  circumference  cuts  a  straight  line  extending  from 
B  toward  D  in  only  one  point;  for  B  is  either  within 
or  on  the  circumference,  since  BC  is  equal  to  or  less 
than  CD.     Therefore,  I  and  D  are  both  at  that  point. 


EQUALITY    OF    TRIANGLES.  97 

Hence,  AI  and   BD   are  equal,  and   the  triangles   are 
equal  (282). 

288.  Corollary — Two  triangles  are  equal  when  they 
have  an  obtuse  or  a  right  angle  in  the  one,  together 
w^ith  the  side  opposite  to  it,  and  one  other  side,  respect- 
ively equal  to  those  parts  in  the  other  triangle  (278). 

The  two  following  are  corollaries  of  the  last  five  theo- 
rems, and  of  the  definition  (40). 

289.  Corollary — In  equal  triangles  each  part  of  one 
is  equal  to  the  corresponding  part  of  the  other. 

290.  Corollary. — In  equal  triangles  the  equal  parts 
are  similarly  arranged,  so  that  equal  angles  are  opposite 
to  equal  sides. 

EXCEPTIONS  TO  THE  GENERAL  RULE. 

201.  A  general  rule  as  to  the  equality  of  triangles 
has  been  given  (281). 
There  are  two  excep- 
tions. 

1.  When  the  three 
angles  are  given. 

For  two  very  unequal  triangles  may  have  the  angles 
of  one  equal  to  those  of  the  other. 

2.  When  two  unequal  sides  and  the  angle  opposite  to 
the  less  are  given. 

For  with   the   sides   AB   and  B 

BC  and  the  angle  A  given, 
there  are  two  triangles,  ABC 
and  ABD. 

292.  The  student  may  show  that  two  parts  alone  are 
never  enough  to  determine  a  triangle. 


98 


ELEMENTS    OF    GEOMETRY. 


UNEQUAL    TRIANGLES. 

29^,  Theorem — Whe7i  two  triangles  have  two  sides  of 
the  one  respectively  equal  to  two  sides  of  the  other,  and  the 
included  angles  unequal,  the  third  side  in  that  triangle 
which  has  the  greater  angle,  is  greater  than  in  the  other. 

Let  BCD  and  AEI  be  two  triangles,  having  BC  equal 
to  AE,  and  BD  equal 
to  AI,  and  the  angle  A 
less  than  B.  Then,  it 
is  to  be  proved  that  CD 
is  greater  than  EI. 

Apply  the  triangle 
AEI  to  BCD,  so  that 
AE  will  coincide  with 
its  equal  BC.  Since  the  angle  A  is  less  than  B,  the 
side  AI  will  fall  within  the  angle  CBD.  Let  BG  be  its 
position,  and  EI  will  fall  upon  CG.  Then  let  a  line  BF 
bisect  the  angle  GBD.     Join  EG. 

The  triangles  GBF  and  BDF  have  the  side  BF  com- 
mon, the  side  GB  equal  to  the  side  DB,  since  each  is 
equal  to  AI,  and  the  included  angles  GBF  and  DBF 
equal  by  construction.  Therefore,  the  triangles  are 
equal  (284),  and  FG  is  equal  to  FD  (289).  Hence,  CD, 
the  sum  of  CF  and  FD,  is  equal  to  the  sum  of  CF  and 
FG  (7),  which  is  greater  than  CG  (54).  Therefore,  CD 
is  greater  than  CG,  or  its  equal  EL 

If  the  point  I  should  fall  within  the  triangle  BCD  or 
on  the  line  CD,  the  demonstration  would  not  be  changed. 

1394.  Theorem. — Conversely,  if  two  triangles  have  tivo 
sides  of  the  one  equal  to  two  sides  of  the  other,  and  the 
third  sides  unequal,  then  the  angles  opposite  the  third  sides 
are  unequal,  and  that  is  greater  which  is  opposite  th<f 
greater  side. 


EQUALITY    OF    TRIANGLES. 


9«.) 


For  if  it  were  less,  then  the  opposite  side  would  be 
les^  (293),  and  if  it  were  equal,  then  the  opposite  sides 
Avould  be  equal  (284) ;  both  of  which  are  contrary  to  the 
hypothesis. 


PROBLEMS    IN    DRAWING. 

29»5.  Problem. —  To    draw   a  triangle  when   the  three 
are  given. 

Let  a,  6,  and  c  be  the  given  lines. 

Draw  the  line  IE  equal  to  ^.    With        ^ 

I  as  a  center,  and  with  the  line  h  as  b 

a  radius,  describe  an  arc,  and  with  E  ^ 

as  a  center  and  the  line  a  as  a  ra- 
dius, describe  a  second  arc,  so  that 
the  two  may  cut  each  other.  Join 
O,  the  point  of  intersection  of  these 
arcs,  with  I  and  with  E.  lOE  is  the 
required  triangle. 

If  c  were  greater  than  the  sum  of  a  and  6,  what  would  have 
been  the  result? 

What,  if  c  were  less  than  the  difference  of  a  and  b  ? 

Has  the  problem  more  than  one  solution ;  that  is,  can  unequal 
triangles  be  drawn  which  comply  with  the  conditions?    Why? 

296.  Corollary. — In  the  same  way,  draw  a  triangle  equal  to  a 
given  triangle. 

207.  Problem. — To  draw  a  triangle,  two  sides  and  the 
included  angle  being  given. 

Let  a  and  b  be  the  given  lines, 
and  E  the  angle. 

Draw  FC  equal  to  b.  At  C  make 
an  angle  equal  to  E.  Take  DC 
equal  to  a,  and  join  FD.  Then  FDC 
is  a  triangle  having  the  required  con- 
ditions. 

Has  this  problem  more  than  one 
solution?     Why? 

Is  this  problem  always  soluble,  whatever  may  be  the  size  o/ 
the  given  angle,  or  the  length  of  the  given  lines?     Why? 


a 


100  ELEMENTS  OF  GEOMETRY. 

29H»  Problem — To  draw  a  triangle  when  one  side  and 
the  adjacent  angles  are  given. 

Let  a  be  tht  given  line,  and  D  and  E  the  angles. 

Draw  BC  equal  to  a.     At  B  make 

a^  angle  equal  to  D,  and  at  C  an  an- 

gle  equal   to   E.     Produce  the  sides 

till  they  meet  at  the  point  F.     FBC  ^y^  ^^^^ 

is  a  triangle   having  the  given   side       :^ "^^ 

and  angles. 

Has  this  problem  more  than  one 
solution  ? 

Can  it  be  solved,  whatever  be  the 
given  angles,  or  the  given  line? 

;S99.  Problem. —  To  draw  a  triangle  when  one  side  and 
two  angles  are  given. 

If  one  of  the  angles  is  opposite  the  given  side,  find  the  sup- 
plement of  the  sum  of  the  given  angles  (214).  This  will  be  the 
other  adjacent  angle  (256).     Then  proceed  as  in  Article  298. 

300.  Problem — To  draiv  a  triangle  when  two  sides 
and  an  angle  opposite  to  one  of  them  are  given. 

Construct  an  angle  equal  to  the  given  angle.  Lay  off  on  one 
side  of  the  angle  the  length  of  the  given  adjacent  side.  With  the 
extremity  of  this  adjacent  side  as  a  center,  and  with  a  radius 
equal  to  the  side  opposite  the  given  angle,  draw  an  arc.  This  arc 
may  cut  the  opposite  side  of  the  angle.  Join  the  point  of  inter- 
section with  the  end  of  the  adjacent  side  which  was  taken  as  a 
center.     A  triangle  thus  formed  has  the  required  conditions. 

The  student  can  better  discuss  this  problem  after  drawing  sev- 
eral triangles  with  various  given  parts.  Let  the  given  angle  vary 
from  very  obtuse  to  very  acute;  and  let  the  opposite  side  vary 
from  being  much  larger  to  much  smaller  than  the  side  adjacent 
to  the  given  angle.  Then  let  the  student  explain  when  this  prob- 
lem has  only  one  solution,  when  it  has  two,  and  when  it  can  not 
be  solved. 

EXERCISES. 

Hot, — 1.  The  base  of  an  isosceles  triangle  is  to  one  of  the 
other  sides  as  three  to  two.  Find  by  construction  and  measure- 
ment, whether  the  vertical  angle  is  acute  or  obtuse. 


SIMILAR   TRIANGLES.  101 

2.  Two  right  angled  triangles  are  equal,  when  any  two  sides  of 
the  one  are  equal  to  the  corresponding  sides  of  the  other. 

3.  Two  right  angled  triangles  are  equal,  when  an  acute  angle 
and  any  side  of  the  one  are  equal  to  the  coiresponding  parts  of 
the  other. 

4.  Divide  a  given  triangle  into  four  equal  parts. 

5.  Construct  a  right  angled  triangle  when, 

I.  An  acute  angle  and  the  adjacent  leg  are  given ; 
II.  An  acute  angle  and  the  opposite  leg  are  given ; 

III.  A  leg  and  the  hypotenuse  are  given  ; 

IV.  When  the  two  legs  are  given. 


SIMILAR    TRIANGLES.    ;      i  v  i    > ' 

302.  Similar  magnitudes  have  been  defined :  to«  fee' 
those  which  have  the  same  form  while  they  differ  in 
extent  (37). 

30S.  Let  the  student  bear  in  mind  that  the  form  of 
a  figure  depends  upon  the  relative  directions  of  its 
points,  and  that  angles  are  differences  in  direction. 
Therefore,  the   definition   may  be  stated  thus : 

Two  figures  are  similar  when  every  angle  that  can  be 
formed  by  lines  joining  points  of  one,  has  its  corre- 
sponding equal  and  similarly  situated  angle  in  the  other. 

ANGLES    EQUAL. 

304.  Theorem. —  Tivo  triangles  are  similar,  when  the 
three  angles  of  the  one  are  respectively  equal  to  the  three 
angles  of  the  other. 

This  may  appear  to  be  only  a  case  of  the  definition 
of  similar  figures ;  but  it  may  be  shown  that  every 
angle  that  can  be  made  by  any  lines  whatever  in  the 
one,  may  have  its  corresponding  equal  and  similarly 
situated  angle  in  the  other. 


102  ELEMENTS    OF   GEOMETRY. 

Let  the  angles  A,  B,  and  C  be  respectively  equal  to 


the  angles  D,  E,  and  F.     Suppose  GH  and  IR  to  be 
any  ;tw.o..  lines  in  the  triangle  ABC. 

'  Join  10  and  GB-     From  F,  the  point  homologous  to 
C,  extend  JL,  making  the  angle  LFE  equal  to  ICB. 

Nusv,  the  triangles  LFE  and  ICB  have  the  angles  B 
and  E  equal,  by  hypothesis,  and  the  angles  at  C  and  F 
equal,  by  construction.  Therefore,  the  third  angles, 
ELF  and  BIC,  are  equal  (262).  By  subtraction,  the 
angles  AIC  and  DLF  are  equal,  and  the  angles  ACI 
and  DEL. 

From  L  extend  LM,  making  the  angle  FLM  equal  to 
CIR.  Then  the  two  triangles  FLM  and  CIR  have  the 
angles  at  C  and  F  equal,  as  just  proved,  and  the  angles 
at  I  and  L  equal,  by  construction.  Therefore,  the  third 
angles,  LMF  and  IRC,  are  equal. 

Join  RG.  Construct  MN  homologous  to  RG,  and  NO 
homologous  to  Gil.  Then  show,  by  reasoning  in  the 
same  manner,  that  the  angles  at  N  are  equal  to  the 
corresponding  angles  at  G ;  and  so  on,  throughout  the 
two  figures. 

The  demonstration  is  similar,  whatever  lines  be  first 
made  in  one  of  the  triangles. 

Therefore,  the  relative  directions  of  all  their  points 
are  the  same  in  both  triangles ;  that  is,  they  have  the 
same  form.     Therefore,  they  are  similar  figures. 


SIMILAR    TRIANGLES. 


103 


305.  Corollary. — Two  similar  triangles  may  be  di- 
vided into  the  same  number  of  triangles  respectively 
similar,  and  similarly  arranged. 

306.  Corollary — Two  triangles  are  similar,  when 
two  angles  of  the  one  are  respectively  equal  to  two 
angles  of  the  other.  For  the  third  angles  must  be 
equal  also  (262). 

307.  Corollary — If  two  sides  of  a 
triangle  be  cut  by  a  line  parallel  to 
the  third  side,  the  triangle  cut  off  is 
similar  to  the  original  triangle  (124). 

308.  Theorem. — Two  triangles  are  similar,  when  the 
sides  of  one  are  parallel  to  those  of  the  other;  or,  when 
the  sides  of  one  are  perpendicular  to  those  of  the  other. 

We  know  (138  and  139)  that  the  angles  formed  by 
lines  which  are  parallel  are  either  equal  or  supplement- 
ary; and  that  the  same  is  true  of  angles  whose  sides 
are  perpendicular  (140).  We  will  show  that  the  angles 
can  not  be  supplementary  in  two  triangles. 

If  even  two  angles  of  one  triangle  could  be  respect- 


ively supplementary  to  two  angles  of  another,  the  sum 
of  these  four  angles  would  be  four  right  angles ;  and 
then  the  sum  of  all  the  angles  of  the  two  triangles 
would  be  more  than  four  right  angles,  which  is  impos- 
sible (255).  Hence, when  two  triangles  have  their  sides 
respectively  parallel  or  perpendicular,  at  least  two  of  the 
angles  of  one  triangle  must  be  equal  to  two  of  the  other. 
Therefore,  the  triangles  are  similar  (306). 


104  ELEMENTS   OF   GEOMETRY. 

SIDES    PROPORTIONAL. 

S09.  Theorem — One  side  of  a  triangle  is  to  the  ho- 
mologous side  of  a  similar  triangle  as  any  side  of  the  first 
is  to  the  homologous  side  of  the  second. 

If  AE  and  BC  are  homologous  sides  of  similar  tri- 

C 

E 


^.. \^ 

Hzl „ \K 


AZ \i  ^^ XD 


angles,  also  EI  and  CD,  then, 

AE  :  BC  :  :  EI  :  CD. 

Take  CF  equal  to  EA,  and  CG  equal  to  EI^  and  join 
FG.  Then  the  triangles  AEI  and  FCG  are  equal  (284), 
and  the  angles  CFG  and  CGF  are  respectively  equal  to 
the  angles  A  and  I,  and  therefore  equal  to  the  angles 
B  and  D.  Hence,  FG  is  parallel  to  BD  (129).  Let  a 
line  extend  through  C  parallel  to  FG  and  BD. 

Suppose  BC  divided  at  the  point  F  into  parts  which 
have  the  ratio  of  two  whole  numbers,  for  example,  four 
and  three.  Then  let  the  line  CF  be  divided  into  four, 
and  BF  into  three  equal  parts.  Let  lines  parallel  to 
BD  extend  from  the  several  points  of  division  till  they 
meet  CD. 

Since  BC  is  divided  into  equal  parts,  the  distances 
between  these  parallels  are  all  equal  (135).  Therefore, 
CD  is  also  divided  into  seven  equal  parts  (134),  of  which 
CG  has  four.     That  is, 

CF  :  CB  :  :  CG  :  CD  :  :  4  :  7. 

But  if  the  lines  BC  and  CF  have  not  the  ratio  of 
two  whole   numbers,  then   let  BC   be  divided  into  any 


SIMILAR   TRIANGLES.  •    105 

number  of  equal  parts,  and  a  line  parallel  to  BD  pass 
through  H,  the  point  of  division  nearest  to  F.  Such  a 
line  must  divide  CD  and  CB  proportionally,  as  just 
proved;  that  is, 

CH  :  CB  : :  CK  :  CD. 

By  increasing  the  number  of  the  equal  parts  into 
which  BC  is  divided,  the  points  H  and  K  may  be  made 
to  approach  within  any  conceivable  distance  of  F  and  G. 
Therefore,  CF  and  CG  are  the  limits  of  those  lines,  CH 
and  CK,  which  are  commensurable  with  BC  and  CD ; 
and  we  may  substitute  CF  and  CG  in  the  last  propor- 
tion for  CH  and  CK. 

Hence,  whatever  be  the  ratio  of  CF  to  CB,  it  is  the 
same  as  that  of  CG  to  CD.  By  substituting  for  CF  and 
CG  the  equal  lines  AE  and  EI,  we  have, 

AE  :  BC  :  :  EI  :  CD. 

By  similar  reasoning  it  may  be  shown  that 
AI  :  BD  :  :  EI  :  CD. 

310.  Corollary. — The  ratio  is  the  same  between  any 
two  homologous  lines  of  two  similar  triangles. 

311.  This  ratio  of  any  side  of  a  triangle  to  the  ho- 
mologous side  of  a  similar  triangle,  is  called  the  linear 
ratio  of  the  two  figures. 

312.  Corollary The  perimeters  of  similar  triangles 

have  the  linear  ratio  of  the  two  figures.     For, 

AE  :  BC  :  :  EI  :  CD  :  :  lA  :  DB. 

Therefore  (23), 

AE+EI+IA  :  BC+CD-f  DB  : :  AE  :  BC. 

313.  Corollary. — If  two  sides  of  a  triangle  are  cut  by 
one  or  more  lines  parallel  to  the  third  side,  the  two  sides 


106 


ELEMENTS   OF    GEOMETRY. 


are   cut    proportionally.     For   the    triangles   so   formed 
are  similar  (807). 

S14.  Corollary — When  several 
parallel  lines  are  cut  by  two  se- 
cants, the  secants  are  divided  pro- 
portionally. 

For  the  secants  being  produced 
till  they  meet,  form  several  simi- 
lar triangles. 

31«>.  Theorem. — Tf  two  sides  of  a  triangle  he  cut  pro- 
portionally hy  a  straight  line,  the  secant  line  is  parallel 
to  the  third  side. 

Let  BCD  be  the  triangle,  and  FG  the  secant. 

A  line  parallel  to  CD  may  pass 
through  F,  and  such  a  line  must 
divide  BD  in  the  same  ratio  as  BC 
(313).  But,  by  hypothesis,  BD  is 
so  divided  at  the  point  G.  There- 
fore, a  line  through  F  parallel  to 
CD,  must  pass  through  G,  and  coin- 
cide with  FG.     Hence,  FG  is  parallel  to  CD. 

316.  Theorem — Two  triangles  are  similar  when  the 
ratios  between  each  side  of  the  one  and  a  corresponding 
side  of  the  other  are  the  same. 

Suppose  AE  :  BC  :  :  EI  :  CD  :  :  AI  :  BD. 

Take  CF  equal  to  EA 
and  CG  equal  to  EI,  and 
join  FG.     Then, 

CF  :  CB  : :  CG  :  CD. 

Therefore,  FG  is  par-      a- 
allel  to  BD  (315),  the  triangles  CFG  and  CBD  are  simi- 
lar (307),  and 

CF  :  CB  :  :  FG  :  BD. 


SIMILAR    TIUANGLES. 


107 


But,  by  hypothesis, 

EA  :  CB  : :  AI  :  BD. 

Hence,  since  CF  is  equal  to  EA,  EG  is  equal  to  AI, 
and  the  triangles  AEI  and  FCG  are  equal.  Therefore, 
the  triangles  AEI  and  BCD  have  their  angles  equal, 
and  are  similar. 

31T.  Theorem. — Two  triangles  are  similar  when  two 
sides  of  the  one  have  respectively  to  two  sides  of  the  other 
the  same  ratio,  and  the  included  angles  are  equal. 

Suppose  AE  :BC  ::  AI  :BD; 

and  let  the  angle  A 
be  equal  to  B. 

Take  BE  equal 
to  AE,  and  BG 
equal  to  AI,  and 
join  EG.  Then  the 
triangles  AEI  and  BEG  are  equal  (284),  and  the  angle 
BEG  is  equal  to  E,  and  BGE  is  equal  to  I.  Since  the 
sides  of  the  triangle  BCD  are  cut  proportionally  by  EG, 
the  angle  BEG  is  equal  to  C,  and  BGE  is  equal  to  D 
(315).  Therefore,  the  triangles  AEI  and  BCD  are  mu- 
tually equiangular  and  similar. 

318.  If  two  similar  triangles  have  two  homologous 
lines  equal,  since  all  other  homologous  lines  have  the 
same  ratio,  they  must  also  be  equal,  and  consequently 
the  two  figures  are  equal.  Thus,  the  equality  of  figures 
may  be  considered  as  a  case  of  similarity. 


PROBLEMS    IN    DRAWING. 

319.  Problem. — To  find  a  fourth  proportional  to  three 
given  straight  lines. 

Let  a  be  the  given  extreme,  and  b  and  c  the  given  means. 
Take  DG  equal  to  a,  the  given  extreme.     Produce  it,  making 


108 


ELEMENTS    OF   GEOMETRY. 


DH   equal    to  c,   one  of  the   means.      From    G  draw  GF  equal 

to  b.     Then,  from  D  draw  a  line  ^ 

through    F,  and  from    H  a  line  ^ 

parallel   to  GF.     Produce  these  ^ 

two  lines   till    they  meet  at  the 

point   K.      FIK    is   the   required 

fourth  proportional. 

For   the   triangles   DGF  and 
DHK  are  similar  (307).    Hence, 

DG  :  GF  :  :  DH  :  HK. 

That  is,     a  :  h  :.  C  :  HK. 

It   is   most   convenient   to   make  GF  and    HK    perpendicular 
to  DH. 

3^0«  Problem.  —  To  divide  a   given    line  into  parts 
having  a  certain  ratio. 

Let  LD  be  the  line  to  be 
divided  into  parts  proportional 
to  the  lines  a,  6,  and  c. 

From  L  draw  the  line  LE 
equal  to  the  sum  of  a,  ^,  and 
c,  making  LF  equal  to  a,  FG 
equal  to  b,  and  GE  equal  to  c. 
Join  DE,  and  draw  GT  and 
FH  parallel  to  DE.  LH,  HI, 
and  ID  are  the  parts  required. 

The  demonstration  is  similar  to  the  last. 

321.  Problem — To  divide  a  given  line  into  any  num- 
ber of  equal  parts. 

This   may  be  done  by  the  last  problem;  but  when   the  given 
line  is  small,  the  following  method  is  preferable. 

To  divide  the  line  AB  into  ten 
equal  parts;  draw  AC  indefinitely, 
and  take  on  it  ten  equal  parts. 
Join  BC,  and  from  the  several 
points  of  division  of  AC,  draw 
lines  parallel  to  AB,  and  produce 

them   to  BC.     The   parallel  nearest  to  AB  is  nine-tenths  of  AB, 
the  next  is  eight-tenths,  and  so  on. 

This  also  dejKnids  upon  similarity  of  triangles. 


SIMILAR    TRIANGLES.  109 

3S!S.  Problem. — To  draw  a  triangle  on  a  given  base, 
similar  to  a  given  triangle. 

Let  this  problem  be  solved  by  the  student. 

IIIGHT    ANGLED    TRIANGLES. 

3!33.  Every  triangle  may  be  divided  into  two  right 
angled  triangles,  by  a  perpendicular  let  fall  from  one  of 
its  vertices  upon  the  opposite  side.  Thus  the  investi- 
gation of  the  properties  of  right  angled  triangles  leads 
to  many  of  the  properties  of  triangles  in  general. 

3S4.  Theorem — If  in  a  rigid  angled  triangle^  a  per- 
pendicular he  let  fall  from  the  vertex  of  the  right  angle 
upon  the  hypotenuse^  then, 

1.  Each  of  the  triangles  thus  formed  is  similar  to  the 
original  triangle; 

2.  Either  leg  of  the  original  triangle  is  a  mean  propor- 
tional between  the  hypotenuse  and  the  adjacent  segment  of 
the  hypotenuse;  and, 

3.  The  perpendicidar  is  a  mean  proportional  between 
the  two  segments  of  the  hypotenuse. 

The  triangles  AEO  and  AEI  have  the  angle  A  com- 
mon, and  the  angles  AEI  and  ^ 
AOE    are    equal,    being    right 
angles.      Therefore,  these   two 
triangles  are  similar  (306) 

That  the  triangles  EOI  and 
EIA  are   similar,  is  proved  by  the  same  reasoning. 

Since  the  triangles  are  similar,  the  homologous  sides 
are  proportional,  and  we  have 

AI  :  AE  ::  AE  :  AO; 

That  is,  the  leg  AE   is  a  mean  proportional  between 


no  ELEMENTS    OF    GEOMETllV. 

the  whole  hypotenuse  and  the  segment  AO  which  h 
adjacent  to  that  leg. 

In  like  manner,  prove  that  EI  is  a  mean  proportional 
between  AI  and  01. 

Lastly,  the  triangles  AEO  and  EIO  are  similar  (304), 
and  therefore, 

AO  :  OE  :  :  OE  :  01. 

That  is,  the  perpendicular  is  a  mean  proportional 
between  the  two  segments  of  the  hypotenuse. 

325.  Corollary. — A   perpendicular  let  fall  from   any 
point  of   a  circumference   upon   a 
diameter,  is  a  mean    proportional 
between  the   two    segments  which 
it  makes  of  the  diameter.  (225) 

32G.  In  the  several  proportions  just  demonstrated, 
in  place  of  the  lines  we  may  substitute  those  numbers 
which  constitute  the  ratios  (14).  Indeed,  it  is  only  upon 
this  supposition  that  the  proportions  have  a  meaning. 
It  is  the  same  whether  these  numbers  be  integers  or 
radicals,  since  we  know  that  the  terms  of  the  ratio  are 
in  fact  numbers. 

327.  Theorem — The  second  power  of  the  length  of 
the  hypotenuse  is  equal  to  the  sum  of  the  second  powers 
of  the  lengths  of  the  two  legs  of  a  right  angled  triayigle. 

Let  h  be  the  hypotenuse,  a  the  perpendicular  let  fall 
upon  it,  h  and  c  the  legs,  and 
d  and  e  the  corresponding  seg- 
ments of  the  hypotenuse  made 
by  the  perpendicular.  That  is, 
these  letters  represent  the  num- 
ber of  times,  whether  integral  or  not,  which  some  unit 
of  length  is  contained  in  each  of  these  lines. 

By  the  second  conclusion  of  the  last  theorem,  we  have 


SIMILAR    TRIANGLES. 


11 


h  :  b  :  :  b  :  d,       and       h  :  c  :  :  c  :  c. 
Hence,  (16),     hd^=b^,  and  he  =  c'^. 

By  adding  these  two,  h  {d-\-  e)  =  b^  +  c^. 


But 


d-\-  e=^h.   Therefore,  It^  =  P  -h 


328.  Theorem. — If,  in  any  triangle,  a  perpendieidar 
be  let  fall  from  one  of  the  vertices  upon  the  opposite  side 
as  a  base,  then  the  whole  base  is  to  the  sum  of  the  other 
two  sides,  as  the  difference  of  those  sides  is  to  the  difference 
of  the  segments  of  the  base. 

Let  a  be  the  perpendicular,  b  the  base,  c  and  d  the 
sides,  and  e  and  i  the 
segments  of  the  base. 

Then,  two  right  an- 
gled triangles  are  formed,  in  one  of  which  we  have 

a^^i'  =  d^; 
and  in  the  other,  a'^-{-e^  =  c'^. 

Subtracting,  i'^ — e'^  =  d'^ — c^. 

Factoring,         (**  +  ^)  {i — e)==(d-\-c)  {d — c). 

Whence  (18),    i  +  e  :  d-\-c  ::  d  —  c  :  i — e. 

329.  Theorem. — If  a  line  bisects  an  angle  of  a  trian- 
gle, it  divides  the  opposite  side  in  the  ratio  of  the  adjacent 
sides. 

If  BF  bisects  the  angle  CBD,  then 
CF  :  FD  :  :  CB  :  BD. 

This  need   be  demonstrated  only  in  the  case  where 
the  sides  adjacent  to  the  bi- 
sected angle  are  not  equal. 

From  C  and  from  D,  let 
perpendiculars  DG  and  CH 
fall  upon  BF,  and  BF  pro-  "''\/ 

duced. 

Then,  the  triangles  BDG  and  BCH  are   similar,  for 


/!•?  ELEMENTS    OF    GEOMETRY. 

thej  have  equal  angles  at  B,  by  hypothesis,  and  at  G 
and  H,  by  construction.     Hence, 

CB  :  BD  :  :  CH  :  DG. 

But  the  triangles  DGF  and  CHF  are  also  mutually 
equiangular  and  similar.     Hence, 

CF  :  FD  : :  CH  :  DG. 

Therefore  (21),     CF  •  FD  :  :  CB  :  BD. 

330.  Problem  in  Drawing. — To  find  a  mean  propor- 
tional to  two  given  straight  lines. 

Make  a  straight  line  equal  to  the  sum  of  the  two.  Upon  this 
as  a  diameter,  describe  a  semi-circumference.  Upon  this  diame- 
ter, erect  a  perpendicular  at  the  point  of  meeting  of  the  two  given 
lines.  Produce  this  to  the  circumference.  The  line  last  drawn 
is  the  required  line. 

Let  the  student  construct  the  figure  and  demonstrate. 

CHORDS,    SECANTS,    AND    TANGENTS 

331.  Theorem. — If  two  chords  of  a  circle  cut  each  other ^ 
the  parts  of  one  may  he  the  extremes,  and  the  parts  of  the 
other  the  means,  of  a  proportion. 

Join  AD  and  CB.  Then  the  two  triangles  AED  and 
CEB  have  the  angle  A  equal  to  „ 

the   angle  C,  since   they  are  in-  ''^^\     ^X 

scribed   in   the   same   arc   (224). 
For  the  same  reason,  the  angles       i 
D  and  B  are  equal.     Therefore, 
the  triangles  are    similar  (306); 
and  we  have  (309),  ]5\^         H.^ 

AE  :  EC  : :  DE  :  EB.  ^ 

332.  Theorem. — If  from  the  same  point,  without  a  cir- 
;le,  two  lines  cutting  the  circumference  extend  to  the  far- 
'her  side,  then  the  whole  of  one  secant  and  its  exterior 


SIMILAR   TRIANGLES. 


113 


part  may  he  the  extremes^  and  the  whole  of  the  other  secant 
and  its  exterior  part  may  he  the  means,  of  a  proportion. 


Joining  BC  and  AD,  the 
triangles  AED  and  CEB  are 
similar;  for  they  have  the 
angle  E  common,  and  the 
angles  at  B  and  D  equal. 
Therefore, 

AE  :  EC  : :  DE  :  EB. 


S33.  Corollary. — If  from  the  same  point  there  be  a 
tangent  and  a  secant,  the  tangent  is  a  mean  propor- 
tional between  the  secant  and  its  exterior  part.  For 
the  tangent  is  the  limit  of  all  the  secants  which  pass 
through  the  point  of  meeting. 

334.  Problem  in  Drawing. — To  divide  a  given  straight 
line  into  two  parts  so  that  one  of  them  is  a  mean  propor- 
tional hettveen  the  whole  line  and  the  other  part. 

This  is  called  dividing  a  line  in  extreme  and  mean  ratio.         -jt^ 
Let  AC   be  the  given   line.     At  C   erect  a   perpendicular,  CT, 
equal  to  half  of  AC.     Join 

A  I.     Take  ID  equal  to  CI,  \e 

and  then  AB  equal  to  AD. 
The  line  AC  is  divided  at 
the  point  B  in  extreme  and 
mean  ratio.     That  is, 
AC  :  AB  :  :  AB  :  BC. 
With  I  as  a  center  and 
IC  as  a  radius,  describe  an  arc  DCE,  and  produce  AI  till  it  meets 
this  arc  at  E.     Then,  AC  is  a  tangent  to  this  arc  (178),  and  there- 
fore (333), 

AE  :  AC  : :  AC  :  AD. 

Or  (24),  AE  — AC  :  AC  : :  AC  — AD  :  AD. 

But  AC  is  twice  IC,  by  construction,  and  DE  is  twice  IC,  be- 
cause DE  is  a  diameter  and  IC  is  a  radius.     Therefore,  the  first 
Gcom.— 10 


/14  ELEMENTS    OF    GEOMETRY. 

term  of  the  last  proportion,  AE  —  AC,  is  equal  to  AE  —  DE,  which 
is  AD;  but  AD  is,  by  construction,  equal  to  AB.  Also,  the  third 
term,  AC  —  AD,  is  equal  to  AC  —  AB,  which  is  BC.  And  the 
fourth  term  is  equal  to  AB.  Substituting  these  equals,  the  pro- 
portion becomes 

AB  :  AC  :  :  BC  :  AB. 
By  inversion  (19),        AC  :  AB  :  :  AB  :  BC. 


ANALYSIS     AND    SYNTHESIS. 

335.  Geometrical  Analysis  is  a  process  employed 
both  for  the  discovery  of  the  solution  of  problems  and 
for  the  investigation  of  the  truth  of  theorems.  Analy- 
sis is  the  reverse  of  synthesis.  Synthesis  commences 
with  certain  principles,  and  proceeds  by  undeniable  and 
successive  inferences.  The  whole  theory  of  geometry 
is  an  example  of  this  method. 

,  In  the  analysis- oi  a  problem,  what  was  required  to 
be  done  is  supposed  to  have  been  effected,  and  the  con- 
sequences are  traced  by  a  series  of  geometrical  con- 
structions and  reasonings,  till  at  length  they  terminate 
in  the  data  of  the  problem,  or  in  some  admitted  truth. 
See'  suggestions,  Article  245. 

In  the  synthesis  of  a  problem,  however,  the  last  con- 
sequence of  the  analysis  is  the  first  step  of  the  process, 
and  the  solution  is  effected  by  proceeding  in  a  contrary 
order  through  the  several  steps  of  the  analysis,  until  the 
process  terminates  in  the  thing  required  to  be  done. 

If,  in  the  analysis,  we  arrive  at  a  consequence  which 
conflicts  with  any  established  principle,  or  which  is  incon- 
sistent with  the  data  of  the  problem,  then  the  solution 
is  impossible.  If,  in  certain  relations  of  the  given  mag- 
nitudes, the  construction  is  possible,  while  in  other  rela- 
tions it  is  impossible,  the  discovery  of  these  relations  is 
a  necessary  part  of  the  discussion  of  the  problem. 


ANALYSIS    AND    SYinTHESIS.  115 

In  the  analysis  of  a  theorem,  the  question  to  be  de- 
termined is,  whether  the  proposition  is  true,  as  stated ; 
.and,  if  so,  how  this  truth  is  to  be  demonstrated.  To  do 
this,  the  truth  is  assumed,  and  the  successive  conse- 
quences of  this  assumption  are  deduced  till  they  term- 
inate in  the  hypothesis  of  the  theorem,  or  in  some 
established  truth. 

The  theorem  will  be  proved  synthetically  by  retracing, 
in  order,  the  steps  of  the  investigation  pursued  in  the 
analysis,  till  they  terminate  in  the  conclusion  which 
had  been  before  assumed.  Tlijs  constitutes  the  demon- 
stration. 

If,  in  the  analysis,  the  assumption  of  the  truth  of  the 
proposition  leads  to  some  consequence  which  conflicts 
with  an  established  principle,  the  false  conclusion  thus 
arrived  at  indicates  the  falsehood  of  the  proposition 
which  was  assumed  to  be  true. 

In  a  word,  analysis  is  used  in  geometry  in  order  to 
discover  truths,  and  synthesis  to  demonstrate  the  truths 
discovered. 

Most  of  the  problems  and  theorems  which  have  been 
given  for  Exercises,  are  of  so  simple  a  character  as 
scarcely  to  admit  of  the  principle  of  geometrical  analy- 
sis being  applied  to  their  solution. 

S36.  A  problem  is  said  to  be  determinate  when  it 
admits  of  one  definite  solution ;  but  when  the  same  con- 
struction may  be  made  on  the  other  side  of  any  given 
line,  it  is  not  considered  a  different  solution.  A  prob- 
lem is  indeterminate  when  it  admits  of  more  than  one 
definite  solution.  Thus,  Article  300  presents  a  case 
where  the  problem  may  be  determinate,  indeterminate, 
or  insolvable,  according  to  the  size  of  the  given  angle 
and  extent  of  the  given  lines. 

The  solution  of  an  indeterminate  problem  frequently 


116  ELEMENTS   OF   GEOMETRY. 

amounts  to  finding  a  geometrical  locus;  as,  to  find  a 
point  equidistant  from  two  given  points ;  or,  to  find  a 
point  at  a  given  distance  from  a  given  line. 


EXERCISES. 

337.  Nearly  all  the  following  exercises  depend  upon 
principles  found  in  this  chapter,  but  a  few  of  them  de- 
pend on  those  of  previous  chapters. 

1.  If  there  be  an  isosceles  and  an  equilateral  triangle  on  the 
same  base,  and  if  the  vertex  of  the  inner  triangle  is  equally  distant 
from  the  vertex  of  the  outer  one  and  from  the  ends  of  the  base, 
then,  according  as  the  isosceles  triangle  is  the  inner  or  the  outer 
one,  its  base  angle  will  be  ^  of,  or  2^  times  the  vertical  angle. 

2.  The  semi-perimeter  of  a  triangle  is  greater  than  any  one  of 
-the  sides,  and  less  than  the  sum  of  any  two. 

3.  Through  a  given  point,  draw  a  line  such  that  the  parts  of 
it,  between  the  given  point  and  perpendiculars  let  fall  on  it  from 
two  other  given  points,  shall  be  equal. 

What  would  be  the  result,  if  the  first  point  were  in  the  straight 
line  joining  the  other  two? 

4.  Of  all  triangles  on  the  same  base,  and  having  their  ver- 
tices in  the  same  line  parallel  to  the  base,  the  isosceles  has  the 
greatest  vertical  angle. 

5.  If,  from  a  point  without  a  circle,  two  tangents  be  made  to 
the  circle,  and  if  a  third  tangent  be  made  at  any  point  of  the  cir- 
<3umference  between  the  first  two,  then,  at  whatever  point  the  last 
tangent  be  made,  the  perimeter  of  the  triangle  formed  by  these 
tangents  is  a  constant  quantity. 

6.  Through  a  given  point  between  two  given  lines,  to  draw  a 
Jine  such  that  the  part  intercepted  by  the  given  lines  shall  be  bi- 
sected at  the  given  point. 

7.  From'a  point  without  two  given  lines,  to  draw  a  line  such 
that  the  part  intercepted  between  the  given  lines  shall  be  equal 
to  the  part  between  the  given  point  and  the  nearest  line. 

8.  The  middle  point  of  a  hypotenuse  is  equally  distant  from 
the  three  vertices  of  a  right  angled  triangle. 


EXERCISES.  117 

9.  Given  one  angle,  a  side  adjacent  to  it,  and  the  difference 
of  the  other  two  sides,  to  construct  the  triangle. 

10.  Given  one  angle,  a  side  opposite  to  it,  and  the  difference  of 
the  other  two  sides,  to  construct  the  triangle. 

11.  Given  one  angle,  a  side  opposite  to  it,  and  the  sum  of  the 
other  two  sides,  to  construct  the  triangle. 

12.  Trisect  a  right  angle. 

13.  If  a  circle  be  inscribed  in  a  right  angled  triangle,  the  dif^ 
ference  between  the  hypotenuse  and  the  sum  of  the  two  legs  is 
equal  to  the  diameter  of  the  circle. 

14.  If  from  a  point  within  an  equilateral  triangle,  a  perpen- 
dicular line  fall  on  each  side,  the  sum  of  these  perpendiculars  is 
a  constant  quantity. 

How  should  this  theorem  be  stated,  if  the  point  were  outside 
of  the  triangle? 

15.  Find  the  locus  of  the  points  such  that  the  sum  of  the  dis- 
tances of  each  from  the  two  sides  of  a  given  angle,  is  equal  to  a 
given  line. 

16.  Find  the  locus  of  the  points  such  that  the  difference  of 
the  distances  of  each  from  two  sides  of  a  given  angle,  is  equal  to 
a  given  line. 

17.  Demonstrate  the  last  corollary  (333)  by  me?ins  of  similar 
triangles. 

18.  To  draw  a  tangent  common  to  two  given  circles. 

19.  To  construct  an  isosceles  triangle,  when  one  side  and  one 
angle  are  given. 

20.  If  in  a  right  angled  triangle  one  of  the  acute  angles  is 
equal  to  twice  the  other,  then  the  Jiypotenuse  is  equal  to  twice  the 
shorter  leg. 

21.  Draw  a  line  DE  parallel  to  the  base  BC  of  a  triangle  ABC, 
so  that  DE  shall  be  equal  to  the  difference  of  BD  and  CE. 

22.  In  a  given  circle,  to  inscribe  a  triangle  similar  to  a  given 
triangle. 

23.  In  a  given  circle,  find  the  locus  of  the  middle  points  of 
those  chords  which  pass  through  a  given  point. 

24.  To  describe  a  circumference  tangent  to  three  given  equal 
circumferences,  which  are  tangent  to  each  other. 


118  ELEMENTS  OF  GEOMETRY. 

25.  If  a  line  bisects  an  exterior  angle  of  a  triangle,  it  divides 
the  base  produced  into  segments 
which    are    proportional   to   the 
adjacent   sides.     That  is,  if  BF 
bisects  the  angle  ABD,  then, 

CF  :   FD  :  :  CB  :  BD. 

26.  The  parts  of  two  parallel  lines  intercepted  by  several  straight 
lines  which  meet  at  one  point,  are  proportional. 

The  converging  lines  are  also  divided  in  tlie  same  ratio. 

27.  Two  triangles  are  similar,  when  two  sides  of  one  are  pro- 
portional to  two  sides  of  the  other,  and  the  angle  opposite  to  that 
side  which  is  equal  to  or  greater  than  the  other  given  side  in  one, 
is  equal  to  the  homologous  angle  in  the  other. 

28.  The  perpendiculars  erected  upon  the  several  sides  of  a  tri- 
angle at  their  centers,  meet  in  one  point. 

29.  The  lines  which  bisect  the  several  angles  of  a  triangle, 
n)eet  in  one  point. 

30.  The  altitudes  of  a  triangle,  that  is,  the  perpendiculars  let 
fall  from  the  several  vertices  on  the  opposite  s'des,  meet  in  one 
pomt. 

31.  The  lines  which  join  the  several  vertices  of  a  triangle  with 
the  centers  of  the  opposite  sides,  meet  in  one  point. 

32.  Each  of  the  lines  last  mentioned  is  divided  at  the  point  of 
meeting  into  two  parts,  one  of  which  is  twice  aa  long  as  the 
otJver. 


QUADKILATERALS.  119 


CHAPTER    VI. 

QUADRILATERALS. 

5538.  In  a  polygon,  two  angles  which  immediately 
succeed  each  other  in  going  round  the  figure,  are  called 
adjacent  angles.  The  student  will  distinguish  adjacent 
angles  of  a  polygon  from  the  adjacent  angles  defined  in 
Article  85. 

A  Diagonal  of  a  polygon  is  a  straight  line  joining 
the  vertices  of  any  two  angles  which 
are  not  adjacent.     Sometimes  a  diag- 
onal is  exterior,  as  the  diagonal  BD 
of  the.  figure  ABCD. 

A  Convex  polygon  has  all  its  di- 
agonals interior. 

A  Concave  polygon  has  at  least  one  diagonal  exte- 
rior, as  in  the  above  diagram. 

Angles,  such  as  BCD,  are  called  reentrant. 

339.  A  Quadrilateral  is  a  polygon  of  four  sides. 

340.  Corollary. — Every  quadrilateral  has  two  diago- 
nals. 

341.  Corollary. — An  interior  diagonal  of  a  quadri- 
lateral divides  the  figure  into  two  triangles. 

EQUAL    quadrilaterals. 

342.  Theorem — Ttvo  quadrilaterals  are  equal  when 
they  are  each  composed  of  two  triangles,  which  are  respect- 
ively equal,  and  similarly  arranged. 


120 


ELEMENTS   OF   GEOxMETRY. 


For,  since  the  parts  are. equal  and  similarly  arranged, 
the  wholes  may  be  made  to  coincide  (40). 

31:3.  Corollary — Conversely,  two  equal  quadrilaterals 
may  be  divided  into  equal  triangles  similarly  arranged. 
In  every  convex  quadrilateral  this  division  may  be  made 
in  either  of  two  ways. 

344.  Theorem. —  Two  quadrilaterals  are  equal  when  the 
four  sides  and  a  diagonal  of  one  are  respectively  equal  to 
the  four  sides  and  the  same  diagonal  of  the  other. 

By  the  same  diagonal  is  meant  the  diagonal  that  has 
the  same  position  with  reference  to  the  equal  sides. 

For,  since  all  their  sides  are  equal,  the  triangles  AEI 


and  BCD  are  equal,  also  the  triangles  AIO  and  BDF 
(282).     Therefore,  the  quadrilaterals  are  equal  (342). 

345.  Theorem Two  quadrilaterals  are  equal   when 

the  four  sides  and  an  angle  of  the  one  are  respectively 
equal  to  the  four  sides  and  the  similarly  situated  angle  of 
the  other. 

By  the    similarly  situated  angle   is  meant  the  angle 
included  by  equal  sides. 

For,  if  the  sides  AE,  IE, 
and  the  included  angle  E 
are  respectively  equal  to 
the  side  BC,  DC,  and  the 
included  angle  C,  then  the 
triangles  AEI  and  BCD  are 
equal  (284) ;    and  AI  is   equal   to  BD.     But  since  the 


QUADRILATERALS.  121 

three  sides  of  the  triangles  AIO  and  BDF  are  respect- 
ively equal,  the  triangles  are  equal  (282).  Hence,  the 
quadrilaterals  are  equal  (342). 

SUM    OF    THE    ANGLES. 

346.  Theorem. — The  sum  of  the  angles  of  a  quadri- 
lateral is  equal  to  four  right  angles. 

For  the  angles  of  the  two  triangles  into  which  every 
quadrilateral  may  be  divided,  are  together  coincident 
with  the  angles  of  the  quadrilateral.  Therefore,  the 
sum  of  the  angles  of  a  quadrilateral  is  twice  the  sum 
of  the  angles  of  a  triangle. 

Let  the  student  illustrate  this  w^ith  a  diagram. 

In  applying  this  theorem  to  a  concave  figure  (338), 
the  value  of  the  reentrant  angle  must  be  taken  on  the 
side  toward  the  polygon,  and  therefore  as  amounting  to 
more  than  two  right  angles. 

INSCRIBED    QUADRILATERAL. 

34T.  Problem. — Any  four  points  of  a  circumference 
may  he  joined  hy  chords,  thus  making  an  inscribed  quad- 
rilateral. 

This  is  a  corollary  of  Article  47. 

348.  Theorem — The  opposite  angles  of  an  inscribed 
quadrilateral  are  supplementary. 

For  the  angle  A  is  measured  by 
half  of  the  arc  EIO  (222),  and  the 
angle  I  by  half  of  the  arc  EAO. 
Therefore,  the  two  together  are 
measured  by  half  of  the  whole  cir- 
cumference, and  their  sum  is  equal 
to  two  right  angles  (207). 
(reoni. — 1 1 


123  ELEMENTS    OF  GEOMETRY. 

TRAPEZOID. 

349.  If  two  adjacent  angles  of  a  quadrilateral  are  sup- 
plemental, the  remaining 
angles    are   also    supple- 
mental (346).     Then,  one 
pair  of  opposite  sides  must     ^  ^ 
be  parallel  (131). 

A  Trapezoid  is  a  quadrilateral  Avhich  has  two  sides 
parallel.     The  parallel  sides  are  called  its  bases. 

350.  Corollary — If  the  angles  adjacent  to  one  base 
of  a  trapezoid  be  equal,  those  adjacent  to  the  other  base 
must  also  be  equal.  For  if  A  and  D  are  equal,  their 
supplements,  B  and  C,  must  be  equal  (96). 

APPLICATION. 

351.  The  figure  described  in  the  last  corollary  is  symmetrical. 
For  it  can  be  divided  into  equal  parts  by 
a    line  joining  the   middle   points  of  the 
bases. 

The  symmetrical  trapezoid  is  used  in 
architecture,  sometimes  for  ornament,  and 
sometimes  as  the  form  of  the  stones  of  an  arch. 

EXERCISES. 

352. — 1.  To  construct  a  quadrilateral  when  the  four  sides  and 
one  diagonal  are  given.  For  example,  the  side  AB,  2  inches;  the 
side  BC,  5;  CD,  3;  DA,  4;  and  the  diagonal  AC,  6  inches. 

2.  To  construct  a  quadrilateral  when  the  four  sides  and  one 
angle  are  given. 

3.  In  a  quadrilateral,  join  any  point  on  one  side  to  each  end  o'^ 
the  side  opposite,  and  with  the  figure  thus  constructed  demonstrate 
the  theorem,  Article  346. 

4.  The  sum  of  two  opposite  sides  of  any  quadrilateral  which  is 


PARALLELOGRAMS.  123 

circumscribed  about  a  circle,  is  equal  to  the  sum  of  the  ether  two 
sides. 

5.  If  the  two  oblique  sides  of  a  trapezoid  be  produced  till  tliey 
meet,  then  the  point  of  meeting,  the  point  of  intersection  of  Hit- 
two  diagonals  of  the  trapezoid,  and  the  middle  points  of  the  two 
bases  are  all  in  one  straight  line. 


PARALLELOGRAMS. 

3«>3.  A  Parallelogram  is  a  quadrilateral  which  has 
its  opposite  sides  parallel. 

354.  Corollary. — Two  adjacent  angles  of  a  parallelo- 
gram are  supplementary.     The 

angles  A  and  B,  being  between 
the  parallels  AD  and  BC,  and 
on  one  side  of  the  secant  AB, 
are  supplementary  (126). 

355.  Corollary. — The  opposite  angles  of  a  parallelo- 
gram are  equal.  For  both  D  and  B  are  supplements  of 
the  angle  C  (96). 

356.  Theorem. — The  opposite  sides  of  a  parallelogram 
are  equal. 

For,  joining  AC  by  a  diagonal,  the  triangles  thus 
formed  have  the  side  AC  common ; 
the  angles  ACB  and  DAC  equal, 
for  they  are  alternate  (125) ;  and 
ACD  and  BAC  equal,  for  the  same 
reason.  Therefore  (285),  the  tri- 
angles are  equal,  and  the  side  AD  ] 
is  equal  to  BC,  and  AB  to  CD. 

357.  Corollary. — When  two  systems 
of  parallels  cross  each  other,  the  parts 
of  one  system  included  between  two 
lines  of  the  other  are  equal. 


124:  ELEMENTiS    OF    (JEOMETRY. 

358.  Corollary. — A  diagonal  divides  a  parallelograTii 
into  two  equal  triangles.  But  the  diagonal  does  not 
divide  the  figure  symmetrical!},  because  the  position  of 
the  sides  of  the  triangles  is  reversed. 

339.  Theorem — If  the  opposite  sides  of  a  quadri- 
lateral are  equal,  the  figure  is  a  parallelogram. 

Join  AC.  Then,  the  triangles  ABC  and  CDA  are 
equal.     For  the  side  AD  is 

equal  to  BO,  and  DC  is  equal  r^ j 

to  AB,  by  hypothesis;  and        p ^ —  q 

they  have  the  side  AC  com- 
mon. Therefore,  the  angles  DAC  and  BCA  are  equal. 
But  these  angles  are  alternate  Avith  reference  to  the 
lines  AD  and  BC,  and  the  secant  AC.  Hence,  AD  and 
BC  are  parallel  (130),  and,  for  a  similar  reason,  AB 
and  DC  are  parallel.  Therefore,  the  figure  is  a  paral- 
lelogram. 

360.  Theorem. — If,  in  a  quadrilateral,  tivo  opposite 
sides  are  equal  and  parallel,  the  figure  is  a  parallelogram. 

If  AD  and  BC  are  both  equal  and  parallel,  then  AB  is 
parallel  to  DC. 

For,  joining  BD,  the  trian-  ^  B 

gles    thus    formed    are    equal,  V  ....■-'"  \ 

■since   they   have  the  side  BD  jp^- q 

common,  the  side  AD  equal  to 

BC,  and  the  angle  ADB  equal  to  its  alternate  DBC  (284). 
Hence,  the  angle  ABD  is  equal  to  BDC.  But  these  are 
alternate  with  reference  to  the  lines  AB  and  DC,  and 
the  secant  BD. 

Therefore,  AB  and  DC  are  parallel,  and  the  figure  is 
a  parallelogram. 

361.  Theorem. — The  diagonals  of  a  parallelogram  bi- 
sect each  other. 


PARALLELOGRAMS.  1*25 

The  diagonals  AC  and  BD  are  each  divided  into  eqiu.l 
parts  at  11,  the  point  ^  B 

of  intersection.  /'^^^X.- 

For     the     triangles         ^^. "       ^^ 

ABH  and  CDH  have       ^  "^ 

the  sides  AB  and  CD  equal  (356),  the  angles  ABH  and 
CDH  equal  (125),  and  the  angles  BAH  and  DCH  equal. 
Therefore,  the  triangles  are  equal  (285),  and  AH  is  equal 
to  CH,  and  BH  to  DH. 

362.  Theorem. — If  the  diagonals  of  a  quadrilateral 
bisect  each  other ^  the  figure  is  a  parallelogram. 

To  be  demonstrated  by  the  student. 

RECTANGLE 

363.  If  one   angle  of  a  parallelogram  is  right,  the 
others  must  be  right  also  (354). 

A  Rectangle  is   a  right  angled 
parallelogram.      The   rectangle   has 
all  the  properties  of  other  parallelo- 
grams, and  the  following  peculiar  to  itself,  which  the 
student  may  demonstrate. 

364.  Theorem — The  diagonals  of  a  rectangle  are  equal. 

RHOMBUS. 

365.  When   two   adjacent   sides   of  a   parallelogram 
are  equal,  all  its  sides  must  be  equal  (356). 

A  Rhombus,  or,  as  sometimes  ^-"^T^v^ 

called,  a  Lozenge,  is  a  parallelo-  ^^-^^     \     ^^>^ 

gram  having  all  its  sides  equal.  ^""^^-v.^^^   j   ^^^-^^^ 

The   rhombus    has   the   follow- 
ing  peculiarities,  which  may   be  demonstrated  by   the 
student. 


126  ELEMENTS    OF    GEOMETRY. 

366.  Theorem. —  The  diagonals  of  a  rhombus  are  per- 
pendicular  to  each  other. 

S67.  Theorem. — The  diagonals  of  a  rhomhui  bisect  its 
angles. 

SQUARE. 

368.  A  Square  is  a  quadrilateral  having  its  sidc3 
equal,  and  its  angles  right  angles.  The  square  may  be 
shown  to  have  ail  the  properties  of  the  parallelogram 
(359),  of  the  rectangle,  and  of  the  rhombus. 

369.  Corollary. — The  rectangle  and  the  square  are 
the  only  parallelograms  which  can  be  inscribed  in  a 
circle  (348). 

EQUALITY. 

370.  Theorem. —  Two  parallelograms  are  equal  when 
two  adjacent  sides  and  the  included  angle  in  the  one,  are 
respectively  equal  to  those  parts  in  the  other. 

For  the  remaining  sides  must  be  equal  (356),  and  this 
becomes  a  case  of  Article  345. 

371.  Corollary. — Two  rectangles  are  equal  when  two 
adjacent  sides  of  the  one,  are  respectively  equal  to  those 
parts  of  the  other. 

373.  Corollary — Two  squares  are  equal  when  a  side 
of  the  one  is  equal  to  a  side  of  the  other. 

APPLICATIONS. 

373.  The  rectangle  is  the  most  frequently  used  of  all  quadri- 
laterals. The  walls  and  floors  of  our  apartments,  doors  and  win- 
dows, books,  paper,  and  many  other  articles,  have  this  form. 

Carpenters  make  an  ingenious  use  of  a  geometrical  principle  in 
order  to  make  their  door  and  window-frames  exactly  rectangular. 
Having  made  the  frame,  with  its  sides  equal  and  its  ends  equal, 


PARALLELOGRAMS?  127 

they  measure  tlie  two  diagonals,  and  make  tlie  frame  take  sucli 
a  shape  that  these  also  will  be  equal. 

In  this  operation,  what  principle  is  applied? 

3*74.  A  rhombus  inscribed  in  a 
rectangle  is  the  basis  of  many  orna- 
ments used  in  arcliitecture  and  other 
work. 

315.  An    instrument  called  parallel  rulers,  used   in    drawin* 
parallel    lines,    consists    of  two  ^ 

rulers,  connected  by  cross  pieces  a  -d 

^/ith  pins   in   their   ends.     The 
rulers  may  turn  upon  the  pins, 
varying  their  distance.     The  dis-        r 
tances  between    the  pins  along  C  D 

tlie  rulers,  that  is,  AB  and  CD, 

must  be  equal;  also,  along  the  cross  pieces,  that  is,  AC  and  BD. 
Then  the  rulers  will  always  be  parallel  to  each  other.  If  one 
ruler  be  held  fast  while  the  other  is  moved,  lines  drawn  along  the 
edge  of  the  other  ruler,  at  difi'erent  positions,  will  be  parallel  to 
each  other. 

What  geometrical  principles  are  involved  in  the  use  of  this 
instrument? 


EXERCISES. 

3*76. — 1.  State  the  converse  of  each  theorem  that  has  been 
given  in  this  chapter,  and  determine  whether  each  of  these  coU' 

verse  propositions  is  true. 

2.  To  construct  a  parallelogram  when  two  adjacent  sides  and 
an  angle  are  given. 

3.  What  parts  need  be  given  for  the  construction  of  a  rect- 
angle? 

4.  What  must  be  given  for  the  construction  of  a  square? 

5.  If  the  four  middle  points  of  the  sides  of  any  quadrilateral 
be  joined  by  straight  lines,  those  lines  form  a  parallelogram. 

6.  If  four  points  be  taken,  one  in  each  side  of  a  square,  at 
equal  distances  from  the  four  vertices,  the  figure  formed  by  join- 
ing these  successive  points  is  a  square. 


12S  ELE^IENTS   OF   GEOMETRY. 

7.  Two  parallelograms  are  similar  wlien  they  have  an  angle  in 
the  one  equal  to  an  angle  in  the  other,  and  these  equal  angles 
included  between  proportional  sides. 


MEASURE    OF    AREA. 

377.  The  standard  figure  for  the  measure  of  surfaces 
is  a  square.  That  is,  the  unit  of  area  is  a  square,  the 
side  of  which  is  the  unit  of  length,  whatever  be  the  ex- 
tent of  the  latter. 

Other  figures  might  be,  and  sometimes  are,  used  for" 
this  purpose;  but  the  square  has  been  almost  univers- 
ally adopted,  because 

1.  Its  form  is  regular  and  simple; 

2.  The  two  dimensions  of  the  square,  its  length  and 
breadth,  are  the  same;  and, 

3.  A  plane  surface  can  be  entirely  covered  with  equal 
squares. 

The  truth  of  the  first  two  reasons  is  already  known 
to  the  student :  that  of  the  last  will  appear  in  the  fol- 
lowing theorem. 

378.  Any  side  of  a  polygon  may  be  taken   as   the 


The  Altitude  of  a  parallelogram  is  the  distance  be- 
tAveen  the  base  and  the  opposite  side.  Hence,  the  alti- 
tude of  a  parallelogram  may  be  taken  in  either  of  two 
ways. 

AREA    OF    RECTANGLES. 

379.  Theovem.-^  The  area  of  a  rectangle  is  measured 
hy  the  product  of  its  base  by  its  altitude. 

That  is,  if  Ave  multiply  the  number  of  units  of  length 
contained  in   the  base,  by  the   number   of   those   units 


MEASURE    OF    AREA.  129 

contained   in  the   altitude,  the   product  is   the  number 
of  units  of  area  contained  in  the  surface. 

Suppose  that  the  base  AB  and  the  altitude  AD  are 
Jiiultiplesof  the  same  unit  of  length, 
for  example,  four  and  three.  Di- 
vide AB  into  four  equal  parts, 
and  through  all  the  points  of  divi- 
sion extend  lines  parallel  to  AD. 
Divide  AD  into  three  equal  parts, 

and  through  the  points  of  division  extend  lines  paral- 
lel to  AB. 

All  the  intercepted  parts  of  these  two  sets  of  parallels 
must  be  equal  (357) ;  and  all  the  angles,  right  angles 
(124).  Thus,  the  whole  rectangle  is  divided  into  equal 
squares  (372).  The  number  of  these  squares  is  equal 
to  the  number  in  one  row  multiplied  by  the  number  of 
rows ;  that  is,  the  number  of  units  of  length  in  the  base 
multiplied  by  the  number  in  the  altitude.  In  the  exam- 
ple taken,  this  is  three  times  four,  or  twelve.  The  result 
would  be  the  same,  whatever  the  number  of  divisions 
in  the  base  and  altitude. 

If  the  base  and  altitude  have  no  common  measure, 
then  we  may  assume  the  unit  of  length  as  small  as  we 
please.  By  taking  for  the  unit  a  less  and  less  part  of 
the  altitude,  the  base  will  be  made  the  limit  of  the  lines 
commensurable  with  the  altitude.  Thus,  the  demonstra-r 
tion  is  made  general. 

380.  Corollary — The  area  of  a  square  is  expressed 
by  the  second  power  of  the  length  of  its  side.  An- 
ciently the  principles  of  arithmetic  were  taught  and  il- 
lustrated by  geometry,  and  we  still  find  the  word  square 
in  common  use  for  the  second  power  of  a  number. 

381.  By  the  method  of  infinites  (203),  the  latter  part 
of  the  above  demonstration  would  consist  in  supposing 


130  ELExMExNTS    OF    GEOMETRY. 

the  base  and  altitude  of  the  rectangle  divided  into  infi- 
nitely small  and  equal  parts ;  and  then  proceeding  to 
form  infinitesimal  squares,  as  in  the  former  part  of  the 
demonstration. 

If  a  straight  line  move  in  a  direction  perpendicular 
to  itself,  it  describes  a  rectangle,  one  of  whose  dimen^ 
sions  is  the  given  line,  and  the  other  is  the  distance 
which  it  has  moved.  Thus,  it  appears  that  the  two  di^ 
mensions  which  every  surface  has  (33),  are  combined  in 
the  simplest  manner  in  the  rectangle. 

A  rectangle  is  said  to  be  coritained  by  its  base  and 
altitude.  Thus,  also,  the  area  of  any  figure  is  called  its 
superficial  contents.  ^ 

APPLICATION. 

3S2.  All  enlightened  nationB  attach  great  importance  to  exact 
and  uniform  standard  measures.  In  this  country  the  standard 
of  length  is  a  yard  measure,  carefully  preserved  by  the  National 
Government,  at  Washington  City.  By  it  all  the  yard  measures 
are  regulated. 

The  standards  generally  used  for  the  measure  of  surface,  are  the 
square  described  upon  a  yard,  a  foot,  a  mile,  or  some  other  cer- 
tain length;  but  the  acre,  one  of  the  most  common  measures  of 
surface,  is  an  exception.  The  number  of  feet,  yards,  or  rods  in 
one  side  of  a  square  acre,  can  only  be  expressed  by  the  aid  of  a 
radical  sign. 

The  public  lands  belonging  to  the  United  States  are  divided 
into  square  townships,  each  containing  thirty-six  square  luiles, 
called  sections. 


AREA    OF    PARALLELOCxHAMS. 

383.   The   area  of  a  parallelogram  is  measured  hy  the 
product  of  its  base  hy  its  altitude. 

At  the  ends  of  the  base  AB  erect  perpen<Iiculars,  and 


MEASURE    OF    AREA. 


131 


produce  them  till  they  meet  the  opposite  side,  in  the 
points  E  and  I . 

Now  the  right  angled  triangles  AED  and  BFC  are 
equal,  having  the  side  BF 
equal  to  AE,  since  they  are 
perpendiculars  between  par- 
allels (133);  and  the  side 
BC  equal  to  AD,  by  hypoth- 
esis (288).     If  each  of  these 

equal  triangles  be  subtracted  from  the  entire  figure, 
ABCE,  the  remainders  ABFE  and  ABCD  must  be 
equivalent.  .But  ABFE  is  a  lectangle  having  the  same 
base  and  altitude  as  the  parallelogram  ABCD.  Hence, 
the  area  of  the  parallelogram  is  measured  by  the  same 
product  as  that  which  measures  t^e  area  of  the  rect- 
angle. 

S84.  Corollary Any  two  parallelograms  have  their 

areas  in  the  same  ratio  as  the  products  of  their  bases 
by  their  altitudes.  Parallelograms  of  equal  altitudes 
have  the  same  ratio  as  their  bases,  and  parallelograms 
of  equal  bases  have  the  same  ratio  as  their  attitudes. 

385.  Corollary. — Two  parallelograms  are  equivalent 
when  they  have  equal  bases  and  altitudes ;  or,  when  the 
two  dimensions  of  the  one  are  the  extremes,  and  the 
two  dimensions  of  the  other  are  the  means,  of  a  pro- 
portion. 

AREA    OF   TRIANGLES. 

#186.  Theorem — The  area  of  a  triangle  is  measured 
hj  half  the  product  of  its  base  hy  its  altitude. 


For  any  triangle  is  one-half  of 
a  parallelogram  having  the  same 
base  and  altitude  (358). 


132  ELEMENTS    OF    GEOMETRY. 

387.  Corollary. — The  areas  of  triangles  are  in  the 
ratio  of  the  products  of  their  bases  by  their  altitudes. 

•$88.  Corollary — Two  triangles  are  equivalent  when 
they  have  equal  bases  and  altitudes. 

389.  Corollary. — If  a  parallelogram  and  a  triangle 
have  equal  bases  and  altitudes,  the  area  of  the  paral- 
lelogram is  double  that  of  the  triangle. 

390.  Theorem If  from  half  the  sum  of  the  three  sides 

of  a  triangle  each  side  be  subtraeted,  and  if  these  remain- 
ders and  the  half  sum  be  multiplied  together,  then  the  square 
root  of  the  product  will  be  the  area  of  the  triqingle. 

Let  DEF  be  any  triangle,  DF  being  the  base  and  EG 
the  altitude.     Let  the  E 

extent  of  the  several 
lines   be   represented 
by  letters ;  that  is,  let 
DF  =  a,  EF-=^),  DE  =  c,  EG  =  A,  QY^m,  DG  =  w,  and 
DE  +  EF  +  FD--S. 

Then  (328),    m^-n\b^c  ::  b—c  :  m—n. 

Therefore,  m  —  n^= — i — . 

'  m-f-n 

By  hypothesis,  m-\-n=^a. 

Adding,  2m=^a-\-- 

Then,  ,  m  = 


a 


za 
Again  (327),  m^^¥  =  b\ 

Substituting  for  m^  its  value,  and  transposing, 


Therefore,  ^  =  Ai*'-(''"^~F- 


iMEASURE    OF    AREA.  133 

But  the  area  of  the  triangle  is  half  the  product  of 
the  base  a  by  the  altitude  A.     Hence, 


area  DEF  =  ~^  =  2V     ~"\ 2a )  * 

In  this  expression,  we  have  the  area  of  the  triangle 
in  terms  of  the  three  sides.  For  greater  facility  of  cal- 
culation it  is  reduced  to  the  following: 


area  =  \i/(a-\-h-^e){a^h  —  c){a  —  h^c){ — a-{-h^c). 

The  exact  equality  of  these  two  expressions  is  shown 
by  performing  as  far  as  is  possible  the  operations  indi- 
cated in  each. 

But,  by  hypothesis,    {a-\-h  -^c)=8  =  2{].\. 
Therefore,  •  {a-\-h—c)  =2/|— A 

and,  (—a^h^c)-=2i^^—a\. 

Substituting  these  in  the  equation  of  the  area,  it  be- 
comes. 


area=^{^  | )( |  — a  )|  \  —  h  )|  \  —  c  ). 

391.  Theorem — The  areas  of  similar  triangles  are  10 
each  other  as  the  squares  of  their  homologous  lines. 


Let  AEI  and  BCD  be  similar  triangles,  and  10  and 
DH  homologous  altitudes. 


134  ELEMENTS    OF    GEOMETRY. 

Then  (310),  10  :  DH  :  :  AE  :  BC. 

Multiply  by  AE  :  BC  :  :  AE  :  BC. 

2        2 

Then,  AE  X  10  :  BC  X  DH  : :  AE  :  BC. 
But  (387), 

AE  X  10  :  BC  X  DH  :  :  area  AEI  :  area  BCD. 
Therefore  (21), 


2 


area  AEI  :  area  BCD  :  :  AE  :  BC. 

In  a  similar  manner,  prove  that  the  areas  have  the 
same  ratio  as  the  squares  of  the  altitudes  10  and  DH, 
or  as  the  squares  of  any  homologous  lines. 

AREA    OF    TRAPEZOIDS. 

392.  Theorem — The  area  of  a  trapezoid  is  equal  to 
half  the  y.  oduct  of  its  altitude  by  the  sum  of  its  parallel 


The  trapezoid  may  be  divided  by  a  diagonal  into  two 
triangles,  having  for  their  bases  the  parallel  sides. 

Thf^  altitude  of  each  of  these  triangles  is  equal  to 
that  of  the  trapezoid  (264).  The  area  of  each  triangle 
being  half  the  product  of  the  common  altitude  by  its 
base,  the  area  of  their  sum,  or  of  the  whole  trapezoid, 
is  half  the  product  of  the  altitude  by  'the  sum  of  the 
bases. 

EXERCISES. 

393. — 1.  Measure  the  length  and  breadth,  and  find  the  area 
of  the  blackboard ;  of  the  floor. 

2.  To  divide  a  given  triangle  into  any  number  of  equivalent 
triangles. 

3.  To  divide  a  given  parallelogram  into  any  number  of  equiva- 
lent parallelograms. 


EQUIVALENT  SURFACES.  135 

4.  To  divide  a  given  trapezoid  into  any  number  of  equivalent 
trapezoids. 

5.  The  area  of  a  triangle  is  equal  to  half  the  product  of  the 
perimeter  by  the  radius  of  the  inscribed  circle. 

6.  What  is  the  radius  of  the   circle  inscribed   in  the  triangle 
whose  sides  are  8,  10,  and  12? 


EQUIVALENT    SURFACES. 

394.  IsoPERiMETRiCAL  figures  are  those  whose  perim- 
eters have  the  same  extent. 

395.  Theorem. — Of  all  equivalent  triangles  of  a  given 
base,  the  one  having  the  least  perimeter  is  isosceles. 

The  equivalent  triangles  having  the  same  base,  AE, 
have    also    the    same    altitude 
(388).      Hence,    their    vertices 
are   in   the   same   line   parallel 
to  the  base,  that  is,  in  DB. 

Now,  the   shortest   line   that        A  E 

can  be  made  from  A  to  E  through  some  point  of  DB, 
will  constitute  the  other  two  sides  of  the  triangle  of 
least  perimeter.  This  shortest  line  is  the  one  making 
equal  angles  with  DB,  as  ACE,  that  is,  making  ACD 
and  ECB  equal  (115).  The  angle  ACD  is  equal  to  its 
alternate  A,  and  the  angle  ECB  to  its  alternate  E 
Therefore,  the  angles  at  the  base  are  equal,  and  the  tri- 
angle is  isosceles. 

396.  Corollary — Of  all  isoperimetrical  triangles  of  ; 
given  base,  the  one  having  the  greatest  area  is  isosceles 

397.  To  draw  a  square  equivalent  to  a  given  fig 
ure,  is  called  the  squaring,  or  quadrature  of  the  figure 
How  this  can  be  done  for  any  rectilinear  figure,  it 
shown  in  the  following. 


136  ELEMENTS   OF   GEOMETRY. 

PROBLEMS    IN    DRAWING. 

308.  Problem. —  To  draw  a  rectangle  ivith  a  given 
base,  equivalent  to  a  given  parallelogram. 

With  the  given  base  as  a  first  term,  and  the  base  and  altitude 
of  the  given  figure  as  the  second  and  third  ternris,  find  a  fourth 
proportional  (319).     This  is  the  required  altitude  (385). 

399.  Problem To   draw   a   square   equivalent   to  a 

given  'parallelogram. 

Find  a  mean  proportional  between  the  base  and  altitude  of  the 
given  figure  (330).     This  is  the  side  of  the  square  (385). 

400.  Problem. —  To  draw  a  triangle  equivalent  to  a 
given  polygon. 

Let  ABCDE  be  the  given  polygon.     Join  DA.     Produce  BA, 
and    through    E    draw   EF 
parallel  to  DA.     Join  DF.  _-^ 

Now,  the  triangles  DAF  fC^'^ //  '''^\ 

and  DAE  are  equivalent,  for  /\      X  /      \  \^Vp 

they  have  the  same  base  DA,  /    y      /  \      \/\ 

and    equal    altitudes,    since  / y    \      /  \      )\  \ 

their  vertices  are  in  the  line  //  \  /  \  /      \\ 

EF  parallel  to  the  base  (264).         |- f g- q 

To  each  of  these  equals,  add 

the  figure  ABCD,  and  we  have  the  quadrilateral  FBCD  equiva- 
lent to  the  polygon  ABCDE.  In  this  manner,  the  number  of 
sides  may  be  diminished  till  a  triangle  is  formed  equivalent  to  the 
given  polygon.     In  this  diagram  it  is  the  triangle  FDG. 

401.  Problem. —  To   draw    a    square   equivalent    to   a 

given  triangle. 

Find  a  mean  proportional  between  the  altitude  and  half  the 
base  of  the  triangle.     This  will  be  the  side  of  the  required  square. 

EQUIVALENT    SQUARES. 

402.  Having  shown  (379)  how  an  area  is  expressed 
by  the  product  of  two  lengths,  it  follows  that  an  equa- 


EQUIVALENT  SURFACES.  137 

tion  will  represent  equivalent  surfaces,  if  each  of  its 
terms  is  composed  of  two  factors  which  represent 
lengths. 

For  example,  let  a  and  h  represent  the  lengths  of  two 
straight  lines.  Now  we  know,  from  algebra,  that  what- 
ever be  the  value  of  a  and  6, 

This  formula,  therefore,  includes  the  following  geomet- 
rical 

403.  Theorem. — The  square  described  upon  the  sum  of 
two  lines  is  equivalent  to  the  sum  of  the  squares  described 
on  the  two  lines,  increased  by  twice  the  rectangle  contained 
by  these  two  lines. 

Since  the   truths  of  algebra  are  universal   in    their 
application,  this  theorem  is  demon- 
strated by  the  truth  of  the   above 
equation. 

Such  a  proof  is  called  algebraic. 
It  is  also  called  analytical,  but  with 
doubtful  propriety. 

Let  the  student  demonstrate  the 
theorem  geometrically,  by  the  aid  of  this  diagram. 

404.  Theorem. — The  square  described  on  the  difference 
of  tivo  straight  lines  is  equivalent  to  the  sum  of  the  squares 
described  on  the  two  lines,  diminished  by  twice  the  rectan- 
gle contained  by  those  lines. 

This  is  a  consequence  of  the  truth  of  the  equation, 

{a  —  bY  =  a''  —  2ab^b\ 

405.  Theorem. —  The  rectangle  contained  by  the  sum 
and  the  difference  of  two  straight  lines  is  equivalent  to  the 
difference  of  the  squares  of  those  lines. 

Geoni.— 12 


ah 

h^ 

d' 

ah 

138  ELEMENTS    OF    GEOMETRY. 

This,  again,  is  proved  by  the  principle  expressed  in 
the  equation, 

{a-^h)  (a  —  h)  =  a''  —  b\ 

406.  These  two  theorems  may  also  be  demonstrated 
by  purely  geometrical  reasoning. 

The  algebraic  method  is  sometimes  called  the  modern, 
\vhile  the  other  is  called  the  ancient  geometry.  The 
algebraic  method  was  invented  by  Descartes,  in  the 
seventeenth  century,  while  the  other  is  twenty  centuries 
older. 

THE    PYTHAGOREAN    THEOREM. 

407.  Since  numerical  equations  represent  geomet- 
rical truths,  the  following  theorem  might  be  inferred 
from  Article  327. 

This  is  called  the  Pythagorean  Theorem,  because  it  was 
discovered  by  Pythagoras.  It  is  also  known  as  the 
Forty-seventh  Proposition,  that  being  its  number  in  the 
First  Book  of  Euclid's  Elements. 

It  has  been  demonstrated  in  a  great  variety  of  ways. 
One  is  by  dividing  the  three  squares  into  parts,  so  that 
the  several  parts  of  the  large  square  are  respectively 
equal  to  the  several  parts  of  the  two  others. 

The  fame  of  this  theorem  makes  it  proper  to  give 
here  the  demonstration  from  Euclid. 

408.  Theorem — The  square  described  on  the  hypote- 
nuse of  a  right  angled  triangle  is  equivalent  to  the  sum 
of  the  squares  described  on  the  tivo  legs. 

Let  ABC  be  a  right  angled  triangle,  having  the  right 
angle  BAG.  The  square  described  on  the  side  BC  is 
equivalent  to  the  sum  of  the  two  squares  described  on 
BA  and  AC.  Through  A  make  AL  parallel  to  BD, 
and  join  AD  and  EC. 


EQUIVALENT  SURFACES. 


139 


Then,  because  each  of  the  angles  BAG  and  BAG  is  a 
right  angle,  the  line 
GAG  is  one  straight 
line  (100).  For  the 
same  reason,  BAH  is 
one  straight  line. 

The  angles  FBG  and 
DBA  are  equal,  since 
each  is  the  sum  of  a 
right  angle  and  the  an- 
gle ABG.  The  two  tri- 
angles FBG  and  DBA 
are  equal,  for  the  side 
FB  in  the  one  is  equal 

to  BA  in  the  other,  and  the  side  BC  in  the  one  is  equal 
to  BD  in  the  other,  and  the  included  angles  are  equal, 
as  just  proved. 

Now,  the  area  of  the  parallelogram  BL  is  double  that 
of  the  triangle  DBA,  because  they  have  the  same  base 
BD,  and  the  same  altitude  DL  (389).  And  the  area  of 
the  square  BG  is  double  that  of  the  triangle  FBG,  be- 
cause these  also  have  the  same  base  BF,  and  the  same 
altitude  FG.  But  doubles  of  equals  are  equal  (7). 
Therefore,  the  parallelogram  BL  and  the  square  BG  are 
equivalent. 

In  the  same  manner,  by  joining  AE  and  BK,  it  is 
demonstrated  that  the  parallelogram  GL  and  the  square 
GH  are  equivalent.  Therefore,  the  whole  square  BE, 
described  on  the  hypotenuse,  is  equivalent  to  the  two 
squares  BG  and  GH,  described  on  the  legs  of  the  right 
angled  triangle. 

409.  Corollary. — The  square  described  on  one  leg  is 
equivalent  to  the  difference  of  the  squares  on  the  hypot- 
enuse and  the  other  leg. 


140 


ELEMENTS   OF   GEOMETRY. 


410.  If  from  the  extremities  of  one  line  perpen- 
diculars be  let  fall  upon  another,  then  the  part  of  the 
second  line  between  the  perpendiculars  is  called  the 
projection  of  the  first  line  on  the  second.  If  one  end 
of  the  first  line  is  in  the  second,  then  only  one  perpen- 
dicular is  necessary. 

411.  Theorem. —  The  square  described  on  the  side  oppo- 
site to  an  acute  angle  of  a  triangle,  is  equivalent  to  the  sum 
of  the  squares  described  on  the  other  two  sides,  diminished 
by  twice  the  rectangle  contained  by  one  of  these  sides  and 
the  projection  of  the  other  on  that  side. 

Let  A  be  the  acute  angle,  and  from  B  let  a  perpen- 
dicular  fall   upon  AC,  produced   if  necessary.     Then, 


AD  is  the  projection  of  AB  upon  AC.  And  it  is  to  be 
proved  that  the  square  on  BC  is  equivalent  to  the  sum 
of  the  squares  on  AB  and  on  AC,  diminished  by  twice 
the  rectangle  contained  by  AC  and  AD. 

2  2 

BD=:AB 


For  (409), 

2  _  2  2 

and  (404),  CD  =  AC  +  AD  —  2AC  X  AD. 

2  _2  2  2 

By  addition,     BD  + CD  =  AB -f  AC— 2AC  X  AD. 


But  the  square  on  BC  is  equivalent  to  BD  -j-  00  (40H). 
Therefore,  it  is  also  equivalent  to 


AB-^AC  — 2ACXAD. 


EQUIVALENT   SURFACES. 


141 


41S.  Theorem — The  square  described  on  the  side  oppo- 
site an  obtuse  angle  of  a  triangle,  is  equivalent  to  the  sum 
of  the  squares  described  on  the  other  two  sides,  increased 
by  twice  the  rectangle  of  one  of  those  sides  and  the  pro- 
jection of  the  other  on  that  side. 

In  the  triangle  ABC,  the  square  on  BC  which  is  op- 
posite the  obtuse  angle  at  B 
A,  is  equivalent  to  the  sum 
of  the  squares  on  AB  and 
on  AC,  and  twice  the  rect- 
angle contained  by  CA  and 
AD. 


For, 


BD-=AB  — AD; 


and  (403), 


CD==AC 

2  2 


AD  +  2ACXAD. 

2 


By  addition,    BD  -f-  CD  =  AB  +  AC  +  2AC  X  AD. 


But, 


BC=-BD  +  CD. 


Therefore,  BC  is  equivalent  to 

2  2 

AB  +  AC  +  2ACXAD. 

413.  Corollary. — If  the  square  described  on  one  side 
of  a  triangle  is  equivalent  to  the  sum  of  the  squares 
described  on  the  other  two  sides,  then  the  opposite  an- 
gle is  a  right  angle.  For  the  last  two  theorems  show 
that  it  can  be  neither  acute  nor  obtuse. 


EXERCISES. 

414. — 1.  When  a  quadrilateral  has  its  opposite  angles  supple- 
mentary, a  circle  can  be  circumscribed  about  it. 

2.  From  a  given  isosceles  triangle,  to  cut  off  a  trapezoid  which 


142  ELEMENTS  OF  GEOMETRY. 

shall  have  the  same  base  as  the  triangle,  and  the  remaining  three 
sides  equal  to  each  other. 

3.  The  lines  which  bisect  the  angles  of  a  parallelogram,  form 
a  rectangle  whose  diagonals  are  parallel  to  the  sides  of  the  paral- 
lelograin. 

4.  In  any  parallelogram,  the  distance  of  one  vertex  from  a 
straight  line  passing  through  the  opposite  vertex,  is  equal  to  the 
sum  or  difference  of  the  distances  of  the  line  from  the  other  two 
vertices,  according  as  the  line  is  without  or  within  the  paral- 
lelogram. 

5.  When  one  diagonal  of  a  quadrilateral  divides  the  figure  into 
equal  triangles,  is  the  figure  necessarily  a  parallelogram? 

6.  Demonstrate  the  theorem.  Article  329,  by  Articles  113  and 
387. 

7.  What  is  the  area  of  a  lot,  which  has  the  shape  of  a  right 
angled  triangle,  the  longest  side  being  100  yards,  and  one  of  the 
other  sides  36  yards. 

8.  Can  every  triangle  be  divided  into  two  equal  parts?  Into 
three?     Into  nine? 

9.  Two  parallelograms  having  the  same  base  and  altituie  are 
equivalent. 

To  be  demonstrated  without  using  Articles  379  or  383. 

10.  A  triangle  is  divided  into  two  equivalent  parts,  by  a  line 
from  the  vertex  to  the  middle  of  the  base. 

To  be  demonstrated  without  the  aid  of  the  principles  of  this 
chapter. 

11.  To  divide  a  triangle  into  two  equivalent  parts,  by  a  line 
drawn  from  a  given  point  in  one  of  the  sides. 

12.  Of  all  equivalent  parallelograms  having  equal  bases,  what 
one  has  the  minimum  perimeter? 

13.  Find  the  locus  of  the  points  such  that  the  sum  of  the 
squares  of  the  distances  of  each  from  two  given  points,  shaU  l-e 
equivalent  to  the  square  of  the  line  joining  the  given  points. 


POLYGONS.  l^:i 


CHAPTER    YII. 

POLYGONS. 

415.  Hitherto  the  student's  attention  has  been  given 
to  polygons  of  three  and  of  four  sides  only.  He  has 
seen  how  the  theories  of  similarity  and  of  linear  ratio 
have  grown  out  of  the  consideration  of  triangles;  and 
how  the  study  of  quadrilaterals  gives  us  the  principles 
for  the  measure  of  surfaces,  and  the  theory  of  equiva- 
lent figures. 

In  the  present  chapter,  some  principles  of  polygons 
of  any  number  of  sides  will  be  established. 

A  Pentagon  is  a  polygon  of  five  sides ;  a  Hexagon 
has  six  sides ;  an  Octagon^  eight ;  a  Decagon,  ten ;  a 
Dodecagon,  twelve;  and  a  Pentedecagon,  fifteen. 

The  following  propositions  on  diagonals,  and  on  the 
sum  of  the  angles,  are  more  general  sta-tements  of  those 
in  Articles  340  to  346. 


DIAGONALS. 

416.  Theorem. — The  number  of  diagonals  from  any 
vertex  of  a  polygon,  is  three  less  than  the  number  of  sides. 

For,  from  each  vertex  a  diagonal  may  extend  to  every 
other  vertex  except  itself,  and  the  one  adjacent  on  each 
side.  Thus,  the  number  is  three  less  than  the  number 
of  vertices,  or  of  sides. 

417.  Corollary. — The   diagonals  from  one  vertex  di- 


144  ELEMENTS   OF   GEOMETRY. 

vide  a  polygon  into  as  many  triangles  as  the  polygon 
has  sides,  less  two. 

Polygons  may  be  divided  into  this  number,  or  into  a 
greater  number  of  triangles,  in  various  ways;  but  a 
polygon  can  not  be  divided  into  a  less  number  of  tri- 
angles than  here  stated. 

418.  Corollary. — The  whole  number  of  diagonals  pos- 
sible in  a  polygon  of  n  sides,  is  J  w  {n  —  3).  For,  if 
we  count  the  diagonals  at  all  the  n  vertices,  we  have 
n  {n —  3),  but  this  is  counting  each  diagonal  at  both 
ends.  This  last  product  must  therefore  be  divided  by 
two. 

EQUAL    POLYGONS. 

419.  Theorem. —  Two  polygons  are  equal  when  they  are 
composed  of  the  same  number  of  triangles  respectively 
equal  and  similarly  arranged. 

This  is  an  immediate  consequence  of  the  definition  of 
equality  (40). 

4!S0.  Corollary — Conversely,  two  equal  polygons  may 
be  divided  into  the  same  number  of  triangles  respect- 
ively equal  and  similarly  arranged. 

421.  Theorem — Two  polygons  are  equal  when  all  the 
sides  and  all  the  diagonals  from  one  vertex  of  the  one,  arc 
respectively  equal  to  the  same  lines  in  the  other,  and  are 
similarly  arranged. 

For  each  triangle  in  the  one  would  have  its  three 
sides  equal  to  the  similarly  situated  triangle  in  the 
other,  and  would  be  equal  to  it  (282).  Therefore,  the 
polygons  would  be  equal  (419). 

422m  Theorem — Two  polygons  are  equal  when  all  the 
sides  and  the  angles  of  the  one  are  respectively  equal  to 
the  same  parts  of  the  other,  and  are  similarly  arranged. 


POLYGONS.  145 

For  each  triangle  in  the  one  is  equal  to  its  homolo- 
gous triangle  in  the  other,  since  they  have  two  sides 
and  the  included  angle  equal. 

It  is  enough  for  the  hypothesis  of  this  theorem,  that 
all  the  angles  except  three  be  among  the  equal  parts. 


SUM    OF    THE    ANGLES. 

423.  Theorem.- — The  sum  of  all  the  angles  of  a  poly- 
gon is  equal  to  twice  as  many  right  angles  as  the  polygon 
has  sides,  less  two. 

For  the  polygon  may  be  divided  into  as  many  trian- 
gles as  it  has  sides,  less  two  (417);  and  the  angles  of 
these  triangles  coincide  altogether  with  those  of  the 
polygon. 

The  sum  of  the  angles  of  each  triangle  is  two  right 
angles.  •  Therefore,  the  sum  of  the  angles  of  the  poly- 
gon is  equal  to  twice  as  many  right  angles  as  it  has 
sides,  less  two. 

The  remark  in  Article  346  applies  as  well  to  this 
theorem. 

424.  Let  R  represent  a  right  angle;  then  the  sum 
of  the  angles  of  a  polygon  of  n  sides  is  2  (n  —  2)  R; 
or,  it  may  be  written  thus,  (2n  —  4)  R. 

The  student  should  illustrate  each  of  the  last  five 
theorems  with  one  or  more  diagrams. 

4!^5.  Theorem. — If  each  side  of  a  convex  polygon  be 
produced,  the  sum  of  all  the  exterior  angles  is  equal  to 
four  right  angles. 

Let  the  sides  be  produced  all  in  one  way;  that  is,  all 
to  the  right  or  all  to  the  left.     Then,  from  any  point  in 
the  plane,  extend  lines  parallel  to  the  sides  thus  pro- 
duced, and  in  the  same  directions. 
Geom. — 13 


146 


ELEMENTS  OF  GEOMETRY. 


The   angles  thus  formed  are  equal  in  number  to  the 
exterior  angles   of  the 
polygon,  and    are    re- 
spectively equal  to  them 
(138).    But  the  sum  of 


,\0.' 


those  formed  about  the      ---V'i 

point  is  equal  to  four 
right  angles  (92). 

Therefore,  the  sum  of  the  exterior  angles  of  the  poly- 
gDn  is  equal  to  four  right  angles. 

426.  This  theorem  will  also  be  true  of  concave  poly- 
gons, if  the  angle  formed  by  producing  one  side  of  the 
reentrant  angle  is  considered  as  a  negative  quantity. 

Thus,  the  remainder,  after 
subtracting  the  angle  formed  at 
A  by  producing  GA,  from  the 
sum  of  the  angles  formed  at 
B,  C,  D,  E,  F,  and  G,  is  four 
right  angles.  This  may  be 
demonstrated  by  the  aid  of 
the  previous  theorem  (423). 


EXEECISES. 

42T. — 1.  What  is  the  number  of  diagonals  that  can  be  in  a 
pentagon?     In  a  decagon? 

2.  What  is  the  sum  of  the  angles  of  a  hexagon?  Of  a  dodec- 
agon ? 

3.  What 'is  the  greatest  number  of  acute  angles  which  a  con- 
vex polygon  can  have? 

4.  Join  any  point  within  a  given  polygon  with  every  vertex  of 
the  polygon,  and  with  the  figure  thus  formed,  demonstrate  the 
theorem,  Article  423. 

5.  Demonstrate  the  theorem,  Article  425,  by  means  of  Article 
23,  and  without  using  Article  92. 


POLYGONS. 


147 


PROBLEMS    IN    DRAWING. 

428.  Problem. — To  draiv  a  polygon  equal  to  a  givem 
polygon. 

By  diagonals  divide  the  given  polygon  into  triangles.  The  prob- 
lem then  consists  in  drawing  triangles  equal  to  given  triangles. 

420.  Problem. —  To  draw  a  polygon  when  all  its  sides 
a7id  all  the  diagonals  from  one  vertex,  are  given  in  their 
proper  order. 

This  consists  in  drawing  triangles  with  sides  equal  to  three 
given  lines  (295). 

430.  Problem — To  draw  a  polygon  when  the  sides 
and  angles  are  given  in  their  order. 

It  is  enough  for  this  problem  if  all  the  angles  except  three  be 
given.     For,  suppose  first  that  the  an- 
gles not  given  are  consecutive,  as  at  D, 
B,  and  C.     Then,  draw  the  triangles  a, 
e,  I,  and  0  (297).    Then,  having  I>€,  com- 
plete the  polygon  by  drawing  the  trian- 
gle  DBC  from    its   three   known   sides 
(295).     Suppose  the  angles  not  given 
were  D,  C,  and  F.     Then,  draw  the  tri- 
angles «,  e,  and  ^,  and   separately,  tJie  triangle  u.     Then,  having 
the  three  sides  of  the  triangle  o,  it  may  be  drawn,  and  the  poly- 
gon completed. 


SIMILAR    POLYGONS. 

431.  Theorem. — Similar  polygons  are  composed  of  the 
same  number  of  triangles,  respectively  similar  and  simi- 
larly arranged. 


Since  the  figures  are  similar,  every  angle  in  one  has 


148  ELEMENTS    OF    GEOMETRY. 

its  corresponding  equal  angle  in  the  other  (303).  If, 
then,  diagonals  be  made  to  divide  one  of  the  polygons 
into  triangles,  every  angle  thus  formed  may  have  its 
corresponding  equal  angle  in  the  other.  Therefore,  the 
triangles  of  one  polygon  are  respectively  similar  to 
those  of  the  other,  and  are  similarly  arranged. 

43!S.  Theorem — If  two  polygons  are  composed  of  the 
same  number  of  triangles  which  are  respectively  similar 
and  are  similarly  arranged,  the  polygons  are  similar. 

By  the  hypothesis,  all  the  angles  formed  by  the  given 
lines  in  one  polygon  have  their  corresponding  equal 
angles  in  the  other.  It  remains  to  be  proved  that  an- 
gles formed  by  any  other  lines  in  the  one  have  their 
corresponding  equal  angles  in  the  other  polygon. 

This  may  be  shown  by  reasoning,  in  the  same  man- 
ner as  in  the  case  of  triangles  (304).  Let  the  student 
make  the  diagrams  and  complete  the  demonstration. 

433.  Theorem. —  Two  polygons  are  similar  when  the 
angles  formed  hy  the  sides  are  respectively  equal,  and  there 
is  the  same  ratio  between  each  side  of  the  one  and  its 
homologous  side  of  the  other. 

Let  all  the  diagonals  possible  extend  from  a  vertex  A 


of  one  polygon,  and  the  same  from  the  homologous  ver- 
tex B  of  the  other  polygon. 

Now  the  triangles  AEI  and  BCD  are  similar,  because 
they  have  two  sides  proportional,  and  the  included  an- 
gles equal  (317). 


SIMILAR    POLYGONS.  149 

Therefore,  EI  :  CD  :  :  AI  :  BD. 

But,  by  hypothesis,  EI  :  CD  :  :  10  :  DF. 
Then  (21),  AI  :  BD  :  :  10  :  DF. 

Also,  if  we  subtract  the  equal  angles  EIA  and  CDB 
from  the  equal  angles  EIO  and  CDF,  the  remainders 
AIO  and  BDF  are  equal.  Hence,  the  triangles  AIO 
and  BDF  are  similar.  In  the  same  manner,  prove  that 
each  of  the  triangles  of  the  first  polygon  is  similar  to 
its  corresponding  triangle  in  the  other.  Therefore,  the 
figures  are  similar  (432). 

As  in  the  case  of  equal  polygons  (422  and  430),  it  is 
only  necessary  to  the  hypothesis  of  this  proposition,  that 
all  the  angles  except  three  in  one  polygon  be  equal  to 
the  homologous  angles  in  the  other. 

4S4.  Theorem. — In  similar  polygons  the  ratio  of  two 
homologous  lines  is  the  same  as  of  any  other  two  homolo- 
gous lines. 

For,  since  the  polygons  are  similar,  the  triangles  which 


compose  them  are  also  similar,  and  (309), 

AE  :  BC  :  :  EI  :  CD  :  :  AI  :  BD  :  :  10  :  DF,  etc. 


This  common  ratio  is  the  linear  ratio  of  the  two 
figures. 

Let  the  student  show  that  the  perpendicular  let  fall 
from  E  upon  OU,  and  the  homologous  line  in  the  other 
polygon,  have  the  linear  ratio  of  the  two  figures. 


(50 


ELEMENTS    OF   GEOMET-RY. 


435.  Theorem — The  perimeters  of  similar  polygons  are 
to  each  other  as  any  two  homologous  lines. 

The  student  may  demonstrate  this  theorem  in  the 
same  manner  as  the  corresponding  propositions  in  trian- 
gles (312). 

436.  Theorem. — The  area  of  any  polygo7i  is  to  the 
area  of  a  similar  polygon,  as  the  square  on  any  line  of  the 
first  is  to  the  square  on  the  homologous  line  of  the  second. 

Let  the  polygons  BCD,  etc.,  and  AEI,  etc.,  be  divided 
into  triangles  by 
homologous  diag- 
onals. The  trian- 
gles thus  formed 
in  the  one  are 
similar  to  those 
formed  in  the 
other  (431). 

Therefore  (391), 

area   BCD  :  area  AEI  :  :  BD  :  AI  :  :  area  BDF  :  area 

2        2  2         2 

AIO  :  :  BF  :  AO  :  :  area  BFG  :  area  AOU  :  :  BG  :  AU 
:  :  area  BGH  :  area  AUY. 

Selecting  from  these  equal  ratios  the  triangles,  area 
BCD  :  area  AEI  : :  area  BDF  :  area  AIO  : :  area  BFG  : 
area  AOU  :  :  area  BGH  :  area  AUY. 

Therefore  (23),  area  BCDFGHB  :  area  AEIOUYA  :  : 

2         2_ 

area  BCD  :  area  AEI ;  or,  as  BC  :  AE ;  or,  as  the  areas 
of  any  other  homologous  parts;  or,  as  the  squares  of 
any  other  homologous  lines. 

437.  Corollary The  superficial  ratio  of  two  similar 

polygons  is  always  the  second  power  of  their  linear 
ratio. 


REGULAR    POLYGONS.  151 

EXERCISES. 

438. — L  Compose  two  polygons  of  the  same  number  of  tri- 
angles respectively  similar,  but  not  similarly  arranged. 

2.  To   draw  a  triangle  similar   to   a  given   triangle,  but  with 
double  the  area. 

3.  What  is  the  relation  between  the  areas  of  the  equilateral  tri- 
angles described  on  the  three  sides  of  a  right  angled  triangle? 


REGULAR    POLYGONS. 

439.  A  Regular  Polygon  is  one  which  has  all  its 
sides  equal,  and  all  its  angles  equal.  The  square  and 
the  equilateral  triangle  are  regular  polygons. 

440.  Theorem — Within  a  regular  'polygon  there  is  a 
point  equally  distant  from  the  vertices  of  all  the  angles. 

Let  ABCD,  etc.,  be  a  regular  polygon,  and  let  lines 
bisecting  the  angles  A  and  B 
extend  till  they  meet  at  0. 
These  lines  will  meet,  for  the 
interior  angles  which  they 
make  with  AB  are  both  acute 
(187). 

In    the    triaui^le   ABO,   the 
angles  at  A  and  B  are  equal,  being  halves  of  the  equal 
angles  of  the  polygon.     Therefore,  the  opposite   sides 
AO  and  BO  are  equal  (275). 

Join  OC.  Now,  the  triangles  ABO  and  BCO  are 
equal,  for  they  have  the  side  AO  of  the  first  equal  to 
BO  of  the  second,  the  side  AB  equal  to  BC,  because  the 
polygon  is  regular,  and  the  included  angles  OAB  and 
OBC  equal,  since  they  are  halves  of  angles  of  the  poly- 
gon.    Hence,  BO  is  equal  to  OC. 

Then,  the  angle  OCB  is  equal  to  OBC  (268),  and  OC 


152  ELEMENTS    OF   GEOMETRY. 

bisects  the  angle  BCD,  which  is  equal  to  ABC.  In  the 
same  manner,  it  is  proved  that  OC  is  equal  to  OD,  and 
so  on.  Therefore,  the  point  0  is  equally  distant  from 
all  the  vertices. 

CIRCUMSCRIBEB    AND    INSCRIBED. 

441.  Corollary — Every  regular  polygon  may  have  a 
circle  circumscribed  about  it.  For,  with  0  as  a  center 
and  OA  as  a  radius,  a  circumference  may  be  described 
passing  through  all  the  vertices  of  the  polygon  (153). 

442.  Theorem — The  point  which  is  equally  distant 
from  the  vertices  is  also  equally  distant  from  the  sides  of 
a  regular  polygon. 

The  triangles  OAB,  OBC,  etc.,  are  all  isosceles.     If 
perpendiculars  be  let  fall  from  0 
upon  the  several  sides  AB,  BC,  ^  5 — ; — S 

etc.,  these  sides  will  be  bisected         A/<  \  \  !  /    ^^D 
(271).     Then,  the  perpendiculars  /     '\\\;/  \ 

will   be   equal,  for  they  will  be  "''4''  ^ 

sides    of    equal    triangles.     But 

they  measure  the  distances  from  0  to  the  several  sides 
of  the  polygon.  Therefore,  the  point  0  is  equally  dis- 
tant from  all  the  sides  of  the  polygon. 

443.  Corollary. — Every  regular  polygon  may  have  a 
circle  inscribed  in  it.  For  with  0  as  a  center  and  OG 
as  a  radius,  a  circumference  may  be  described  passing 
through  the  feet  of  all  these  perpendiculars,  and  tangent 
to  all  the  sides  of  the  polygon  (178) ,  and  therefore  in- 
scribed in  it  (253). 

444.  Corollary. — A  regular  polygon  is  a  symmetrical 
figure. 

445.  The  center  of  the  circumscribed  or  inscribed  cir- 
cle is  also  called  the  center  of  a  regidar  polygon.     The 


REGULAR    POLYGONS. 


153 


radius  of   the   circumscribed   circle   is   also   called   the 
radius  of  a  regular  polygon. 

The  Apothem  of  a  regular  polygon  is  the  radius  of 
the  inscribed  circle. 

446.  Theorem — If  the  circumference  of  a  circle  he  di- 
vided into  equal  arcs,  the  chords  of  those  equal  arcs  will 
he  the  sides  of  a  regular  polygon. 

For  the  sides  are  all  equal,  being  the  chords  of  equal 
arcs  (185);  and  the  angles  are  all  equal,  being  inscribed 
in  equal  arcs  (224). 

447.  Corollary — An  angle  formed  at  the  center  of  a 
regular  polygon  by  lines  from  adjacent  vertices,  is  an 
aliquot  part  of  four  right  angles,  being  the  quotient  of 
four  right  angles  divided  by  the  number  of  the  sides  of 
the  polygon. 

448.  Theorem — If  a  circumference  he  divided  into 
equal  arcs,  and  lines  tangent  at  the  several  points  of  divi- 
sion he  produced  until  they  meet,  these  tangents  are  the 
sides  of  a  regular  polygon. 

Let  A,  B,  C,  etc.,  be  points  of  division,  and  F,  D,  and 
E    points   where    the   tangents 
meet. 

Join  GA,  AB,  and  BC. 

Now,  the  triangles  GAF, 
ABD,  and  BCE  have  the  sides 
GA,  AB,  and  BC  equal,  as  they 
are  chords  of  equal  arcs;  and 
the  angles  at  G,  A,  B,  and  C 
equal,  for  each  is  formed  by  a 
tangent  and  chord  which  inter- 
cept equal  arcs  (226).  Therefore,  these  triangles  are 
all  isosceles  (275),  and  all  equal  (285);  and  the  angles 
F,    D,    and   E    are    equal.     Also,   FD    and    DE,  being 


154  ELEMENTS    OF    GEOMETRY. 

doubles  of  equals,  are  equal.  In  the  same  manner,  it  is 
proved  that  all  the  angles  of  the  polygon  FDE,  etc.,  are 
equal,  and  that  all  its  sides  are  equal.  Therefore,  it  is 
a  regular  polygon. 

REGULAR    POLYGONS    SIMILAR. 

449.  Theorem. — Regular  polygons  of  the  same  number 
of  sides  are  similar. 

Since  the  polygons  have  the  same  number  of  sides, 
the  sum  of  all  the  angles  of  the  one  is  equal  to  the  sum 
of  all  the  angles  of  the  other  (423).  But  all  the  angles 
of  a  regular  polygon  are  equal  (439).  Dividing  the 
equal  sums  by  the  number  of  angles  (7),  it  follows  that 
an  angle  of  the  one  polygon  is  equal  to  an  angle  of 
the  other. 

Again :  all  the  sides  of  a  regular  polygon  are  equal. 
Hence,  there  is  the  same  ratio  between  a  side  of  the 
first  and  a  side  of  the  second,  as  between  any  other  side 
of  the  first  and  a  corresponding  side  of  the  second. 
Therefore,  the  polygons  are  similar  (433). 

450.  Corollary — The  areas  of  two  regular  polygons 
of  the  same  number  of  sides  are  to  each  other  as  the 
squares  of  their  homologous  lines  (436). 

451.  Corollary. — The  ratio  of  the  radius  to  the  side 
of  a  regular  polygon  of  a  given  number  of  sides,  is  a 
constant  quantity.  For  a  radius  of  one  is  to  a  radius 
of  any  other,  as  a  side  of  the  one  is  to  a  side  of  the 
other  (434).  Then,  by  alternation  (19),  the  radius  is  to 
the  side  of  one  regular  polygon,  as  the  radius  is  to  the 
side  of  any  other  regular  polygon  of  the  same  number 
of  sides. 

452.  Corollary. — The  same  is  true  of  the  apothem  and 
side,  or  of  the  apothem  and  radius. 


REGULAR    POLYGONS. 


155 


PROBLEMS    IN     DRAWING. 


453.  Problem. — To  inscribe  a  square  in  a  given  circle. 

Draw  two  diameters  perpendicular  to  each  other.  Join  their 
extremities  by  chords.     These  chords  form  an  inscribed  square. 

For  the  angles  at  the  center  are  equal  by  construction  (90). 
Therefore,  their  intercepted  arcs  are  equal  (197),  and  the  chords 
of  those  arcs  are  the  sides  of  a  regular  polygon  (446). 

454.  Problem. — To  inscribe  a  regular  hexagon  in  a 
circle. 

Suppose  the  problem  solved  and    the   figure  completed.     Join 
two  adjacent  angles  with  the  center, 
making  the  triangle  ABC. 

Now,  the  angle  C,  being  measured 
by  one-sixth  of  the  circumference,  is 
equal  to  one-sixth  of  four  right  an- 
gles, or  one-third  of  two  right  an- 
gles. Hence,  the  sum  of  the  two 
angles,  CAB  and  CBA,  is  two-thirds 
of  two  right  angles  (256).  But  CA 
and  CB  are  equal,  being  radii;  there- 
fore, the  angles  CAB  and  CBA  are  equal  (268),  and  each  of  them 
must  be  one-third  of  two  right  angles.  Then,  the  triangle  ABC, 
being  equiangular,  is  equilateral  (276).  Therefore,  the  side  of  an 
inscribed  regular  hexagon  is  equal  to  the  radius  of  the  circle. 

The  solution  of  the  problem  is  now  evident — apply  the  radius 
to  the  circumference  six  times  as  a  chord. 

455.  Corollary. — Joining  the  alternate  vertices  makes  an  in- 
scribed equilateral  triangle. 

456.  Problem. — To  inscribe  a  regular  decagon  in  a 
given  circle. 

Divide  the  radius  CA  in  extreme  and  mean  ratio,  at  the  point 
B.  \334)  BC  is  equal  to  the  side  of  a  regular  inscribed  decagon. 
That  is,  if  we  apply  BC  as  a  chord,  its  arc  will  be  one-tenth  of 
the  whole  circumference. 

Take  AD,  making  the  chord  AD  equal  to  BC.  Then  join  DC 
and  DB. 

Then,  by  construction,  CA  :  CB  :  :  CB  :  BA. 


15G  ELEMENTS   OF   GEOMETRY. 

Substituting  for  CB  its  equal  DA, 

CA  :  DA  ::  DA*  BA. 

Then  the  triangles  CDA  and  BDA  are  similar,  for  they  have 
those  sides  proportional  which  include 
tlie  common  angle  A  (317).     But  the 
triangle  CDA  being  isosceles,  the  tri- 
ano-le  BDA  is  the  same.     Hence,  DB 

is    equal    to    DA,    and    also    to    BC.        \  ^  B     j 

Therefore,  the  angle  C  is  equal  to  the 
angle  BDC  (26S).  But  it  is  also  equal 
to  BDA.  It  follows  that  the  angle 
CDA  is  twice  the  angle  C.  The  angle  at  A  being  equal  to  CDA, 
the  angle  C  must  be  one-fifth  of  the  sum  of  these  three  angles; 
that  is,  one-fifth  of  two  right  angles  (255),  or  one-tenth  of  four 
right  angles.  Therefore,  the  arc  AD  is  one-tenth  of  the  circum- 
ference (207);  and  the  chord  AD  is  equal  to  the  side  of  an  in- 
scribed regular  decagon. 

45T, — Corollary, — By  joining  the  alternate  vertices  of  a  deca- 
gon, we  may  inscribe  a  regular  pentagon. 

458.  Corollary. — A  regular  pentedecagon,  or  polygon  of  fifteen 
sides,  may  be  inscribed,  by  subtracting  the  arc  subtended  by  the 
side  of  a  regular  decagon  from  the  arc  subtended  by  the  side  of 
a  regular  hexagon.  The  remainder  is  one-fifteenth  of  the  circum- 
ference, for  ^  — to  =  tV 

459.  Problem. — Given  a  regular  polygon  inscribed  in 
a  circle^  to  inscribe  a  regular  polygon  of  double  the  num- 
ber of  sides. 

Divide  eich  arc  subtended  by  a  given  side  into  two  equal  parts 
(194).  Join  the  successive  points  into  which  the  circumference  is 
divided.     Thj  figure  thus  formed  is  the  required  polygon. 

480.  We  have  now  learned  how  to  inscribe  regular  polygons 
of  3,  4,  5,  and  15  sides,  and  of  any  number  that  may  arise  from 
doubling  either  of  these  four. 

The  problem,  to  inscribe  a  regular  pol3'gon  in  a  circle  by 
means  of  straight  lines  and  arcs  of  circles,  can  be  solved  in  only 
a  limited  number  of  cases.  It  is  evident  that  the  solution  depends 
upon  the  division  of  the  circumference  into  any  number  of  equal 
parts;  and  this  depends  upon  the  division  of  the  sum  of  four  right 
angles  into  aliquot  parts. 


REGULAR    POLYGONS.  157 

461.  Notice  that  the   regular  decagon  was  drawn  by  the  aid 

of  two  isosceles  triangles  composing 
a  third,  one  of  the  two  being  simi- 
lar to  the  whole.  Now,  if  we  could 
combine  three  isosceles  triangles  in 
this  manner,  we  could  draw  a  regu- 
lar polygon  of  fourteen,  and  then 
one  of  seven    sides. 

However,  this  can  not  be  done  by  means  only  of  straight  lines 
and  arcs  of  circles. 

The  regular  polygon  of  seventeen  sides  has  been  drawn  in  more 
tlian  one  way,  using  only  straight  lines  and  arcs  of  circles.  It 
has  also  been  shown,  that  by  the  same  means  a  regular  polygon 
of  two  hundred  and  fifty-seven  sides  may  be  drawn.  No  others 
are  known  where  the  number  of  the  sides  is  a  prime  number. 

403.  Problem — Given  a  regular  polygon  inscribed  in 
a  circle,  to  circumscribe  a  similar  polygon. 

The  vertices  of  the  given  polygon  divide  the  circumference  into 
equal  parts.  Through  these  points  draw  tangents.  These  tan- 
gents produced  till  they  meet,  form  the  required  polygon  (448), 


EXERCISES. 

463. — 1.  First  in  right  angles,  and  then  in  degrees,  express 
the  value  of  an  angle  of  each  regular  polygon,  from  three  sides 
up  to  twenty. 

2.  First  in  right  angles,  and  then  in  degrees,  express  the  value 
of  an  angle  at  the  center,  subtended  by  one  side  of  each  of  the 
same  polygons. 

3.  To  construct  a  regular  octagon  of  a  given  side. 

4.  To  circumscribe  a  circle  about  a  regular  polygon. 

5.  To  inscribe  a  circle  in  a  regular  polygon. 

6.  Given  a  regular  inscribed  polygon,  to  circumscribe  a  similar 
polygon  whose  sides  are  parallel  to  the  former. 


7.  The   diagonal   of  a  square   is   to  its 
side  as  the  square  root  of  2  is  to  1, 


1 56 


ELEMENTS    OF    GEOMETRY. 


A    PLANE    OF    REGULAR    POLYGONS. 

461.  In  order  that  any  plane  surface  may  be  entirely 
covered  by  equal  polygons,  it  is  necessary  that  the  fig- 
ures be  such,  and  such  only,  that  the  sum  of  three  or 
more  of  their  angles  is  equal  to  four  right  angles  (92). 

Hence,  to  find  what  regular  polygons  will  fit  together 
so  as  to  cover  any  plane  surface,  take  them  in  order 
according  to  the  number  of  their  sides. 

Each  angle  of  an  equilateral  triangle 
is  equal  to  one-third  of  two  right  an- 
gles. Therefore,  six  such  angles  ex- 
actly make  up  four  right  angles;  and 
the  equilateral  triangle  is  such  a  fig- 
ure as  is  required. 

465.  Each  angle  of  the  square  is  a  right  angle,  four 
of  which   make   four  right  angles.     So 

that  a  plane  can  be  covered  by  equal 
squares. 

One   angle  of  a  regular  pentagon  is 
the  fifth  part  of  six  right  angles.     Three 
of  these  are  less  than,  and  four  exceed 
four  right  angles;  so  that  the  regular  pentagon  is  not 
such  a  figure  as  is  required. 

466.  Each  angle  of  a  regular  hexagon  is  one-sixth 
of  eight  right  angles.     Three  such  make 

up  four  right  angles.  Hence,  a  plane 
may  be  covered  with  equal  regular  hexa- 
gons. This  combination  is  remarkable  as 
being  the  one  adopted  by  bees  in  form- 
ing the  honeycomb. 

4G7.  Since  each  angle  of  a  regular  polygon  evi- 
dently increases  when  the  number  of  sides  increases, 
and  since  three  angles  of  a  regular  hexagon  are   equal 


ISOPEKIMETRY.  159 

to  i(/  r  right  angles,  therefore,  three  angles  of  any  reg- 
ular polygon  of  more  than  six  sides,  must  exceed  four 
right  angles. 

Hence,  no  other  regular  figures  exist  for  the  purpose 
here  required,  except  the  equilateral  triangle,  the  square, 
and  the  reguliir  hsxagon. 


ISOPEKIMETKY. 

46S.  Theorem — Of  all  equivalent  polygons  of  the  same 
number  of  sides,  the  one  having  the  least  perimeter  is  reg- 
ular. 

Of  several  equivalent  polygons,  suppose  AB  and  BO 
to  be  two  adjacent  sides  of 
the  one   having  the  least  .^J^-- 

perimeter.      It    is    to    be  k,^*'^'^^     ^^^^r^ 

proved,   first,   that    these  /  A 

sides  are  equal.  /  \ 

Join  AC.     Now,  if  AB       /  \ 

and    BC   were   not  equal, 

there  could  be  constructed  on  the  base  AC  an  isosceles 
triangle  equivalent  to  ABC,  whose  sides  would  have  less 
extent  (395).  Then,  this  new  triangle,  with  the  rest  of 
the  polygon,  would  be  equivalent  to  the  given  polygon, 
and  have  a  less  perimeter,  which  is  contrary  to  the 
hypothesis. 

It  follows  that  AB  and  BC  must  be  equal.  So  of 
every  two  adjacent  sides.  Therefore,  the  polygon  is 
c({uilateral. 

It  remains  to  be  proved  that  the  polygon  will  have 
all  its  angles  equal. 

Suppose  AB,  BC,  and  CD  to  be  adjacent  sides. 
Produce  AB  and  CD  till  they  meet  at  E.  Now  the 
triangle  BCE  is  isosceles.     For  if  EC,  for  example,  were 


160  ELEMENTS   OF    GEOMETRY. 

longer  than  EB,  we  could  then  take  EI  equal  to  EB,  and 

EF  equal  to  EC,  and  we 

could  join  FI,  making  the  ^S 

two    triangles    EBC    and 

EIF  equal  (284). 

Then,  the  new  polygon, 
having  AFID  for  part  of 
its    perimeter,    would    be 

equivalent  and  isoperimetrical  to  the  given  polygon  hav- 
ing ABCD  as  part  of  its  perimeter.  But  the  given 
polygon  has,  by  hypothesis,  the  least  possible  perimeter, 
and,  as  just  proved,  its  sides  AB,  BC,  and  CD  are  equal. 

If  the  new  polygon  has  the  same  area  and  perime- 
ter, its  sides  also,  for  the  same  reason,  must  be  equal ; 
that  is,  AF,  FI,  and  ID.  But  this  is  absurd,  for  AF  is 
less  than  AB,  and  ID  is  greater  than  CD.  Therefore, 
the  supposition  that  EC  is  greater  than  EB,  which  sup- 
position led  to  this  conclusion,  is  false.  Hence,  EB  and 
EC  must  be  equal. 

Therefore,  the  angles  EBC  and  ECB  are  equal  (268), 
and  their  supplements  ABC  and  BCD  are  equal.  Thus, 
it  may  be  shown  that  every  two  adjacent  angles  are 
equal. 

It  being  proved  that  the  polygon  has  its  sides  equal 
and  its  angles  equal,  it  is  regular. 

469.  Corollary. — Of  all  isoperimetrical  polygons  of 
the  same  number  of  sides,  that  which  is  regular  has  the 
greatest  area. 

470.  Theorem. — Of  all  regular  equivalent  polygons^ 
that  which  has  the  greatest  number  of  sid-es  has  the  least 
perimeter. 

It  will  be  sufficient  to  demonstrate  the  principle,  when 
one  of  the  equivalent  polygons  has  one  side  more  than 
ihe  other. 


ISOPERIMETRY. 


161 


In  the  polygon  having  the  less  number  of  sides,  join 
the  vertex  C  to  any  point,  as  H,  of  the  side  BG.     Then, 


01  CH  construct  an  isosceles  triangle,  CKH,  equivalent 
to  CBH. 

Then  HK  and  KC  are  less  than  HB  and  BC ;  there- 
fore, the  perimeter  GHKCDF  is  less  than  the  perimeter 
of  its  equivalent  polygon  GBCDF.  But  the  perimeter 
of  the  regular  polygon  AO  is  less  than  the  perimeter  of 
its  equivalent  irregular  polygon  of  the  same  number  of 
sides,  GHKCDF  (468).  So  much  more  is  it  less  than 
the  perimeter  of  GBCDF. 

471.  Corollary. — Of  two  regular  isoperimetrical  poly- 
gons, the  greater  is  that  which  has  the  greater  number 
of  sides. 

EXERCISES. 


472. — 1.  Find  the  ratios  between  the  side,  the  radius,  and  the 
{\pothem,  of  the  regular  polygons  of  three,  four,  five,  six,  and  eiglit 
sides, 

2.  If  from  any  point  within  a  given  regular  polygon,  perpen- 
diculars be  let  fall  on  all  the  sides,  the  sum  of  these  perpendicu- 
lars is  a  constant  quantity. 

3.  If  from  all  the  vertices  of  a  regular  polygon,  perpendiculars 
be  let  fall  on  a  straight  line  which  passes  through  its  center,  the 

Geom. — 14 


162  ELEMENTS  OF  GEOMETRY. 

sum  of  the  perpendiculars  on  one  side  of  this  line  is  equal  to  the 
sum  of  those  on  the  other. 

4.  If  a  regular  pentagon,  hexagon,  and  decagon  be  inscribed 
in  a  circle,  a  triangle  having  its  sides  respectively  equal  to  the 
sides  of  these  three  polygons  will  be  right  angled. 

5.  If  two  diagonals  of  a  regular  pentagon  cut  each  other,  each 
is  divided  in  extreme  and  mean  ratio. 

6.  Three  houses  are  built  with  walls  of  the  same  aggregate 
length;  the  first  in  the  shape  of  a  square,  the  second  of  a  rectan- 
gle, and  the  third  of  a  regular  octagon.  Which  has  the  greatest 
amount  of  room,  and  which  the  least? 

7.  Of  all  triangles  having  two  sides  respectively  equal  to  two 
given  lines,  the  greatest  is  that  where  the  angle  included  between 
the  given  sides  is  a  right  angle. 

8.  In  order  to  cover  a  pavement  with  equal  blocks,  in  the  shape 
of  regular  polygons  of  a  given  area,  of  what  shape  must  they  be 
that  the  entire  extent  of  the  lines  between  the  blocks  shall  be  a 
minimum. 

9.  All  the  diagonals  being  formed  in  a  regular  pentagon,  the 
figure  inclosed  by  them  is  a  regular  pentagon. 


CIRCLES. 


133 


CHAPTER    VIII. 


CIRCLES. 


473.  The  properties  of  the  curve  which  bounds  a 
circle,  and  of  some  straight  lines  connected  with  it, 
were  discussed  in  a  former  chapter.  Having  now  learned 
the  properties  of  polygons,  or  rectilinear  figures  inclos- 
ing a  plane  surface,  the  student  is  prepared  for  the 
study  of  the  circle  as  a  figure  inclosing  a  surface. 

The  circle  is  the  only  curvilinear  figure  treated  of 
in  Elementary  Geometry.  Its  discussion  will  complete 
this  portion  of  the  work.  The  properties  of  other 
curves,  such  as  the  ellipse  which  is  the  figure  of  the 
orbits  of  the  planets,  are  usually  investigated  by  the 
application  of  algebra  to  geometry. 

474.  A  Segment  of  a  circle  is  that  portion  cut  oif 
by  a  secant  or  a  chord.  Thus,  ABC  and  CDE  are  seg- 
ments. 

D 


A  Sector  of  a  circle  is  that  portion  included  between 
two  radii  and  the  arc  intercepted  by  them.  Thus,  GHI 
is  a  sector. 


164  ELEMENTS    OF  C.KOMETRY. 

THE    LIMIT    OF    INSCRIBED    POLYGONS. 

47*5.  Theorem — A  circle  is  the  limit  of  the  polygons 
which  can  be  inscribed  in  it,  also  of  those  which  can  be 
circumscribed  about  it. 

Having  a  polygon  inscribed  in  a  circle,  a  second  poly- 
gon may  be  inscribed  of  double  the  number  of  sides. 
Then,  a  polygon  of  double  the  number  of  sides  of  the 
second  may  be  inscribed,  and  the  process  repeated  at 
will. 

Let  the  student  draw  a  diagram,  beginning  with  an 
inscribed  square  or  equilateral  triangle.  Very  soon  the 
many  sides  of  the  polygon  become  confused  with  the 
circumference.  Suppose  we  begin  with  a  circumscribed 
regular  polygon;  here,  also,  we  may  circumscribe  a 
regular  polygon  of  double  the  number  of  sides.  By 
repeating  the  process  a  few  times,  the  polygon  become?: 
inseparable  from  the  circumference. 

The  mental  process  is  not  subject  to  the  same  limits 
that  we  meet  with  in  drawing  the  diagrams.  We  may 
conceive  the  number  of  sides  to  go  on  increasing  to  any 
number  whatever.  At  each  step  the  inscribed  polygon 
grows  larger  and  the  circumscribed  grows  smaller,  both 
becoming  more  nearly  identical  with  the  circle. 

Now,  it  is  evident  that  by  the  process  described,  the 
polygons  can  be  made  to  approach  as  nearly  as  we  please 
to  equality  with  the  circle  (35  and  36),  but  can  never  en- 
tirely reach  it.  The  circle  is  therefore  the  limit  of  the 
polygons  (198). 

476.  Corollary. — A  circl'e  is  the  limit  of  all  regular 
polygons  whose  radii  are  equal  to  its  radius.  It  is  also 
the  limit  of  all  regular  polygons  whose  apothems  are 
equal  to  its  radius.  The  circumference  is  the  limit  of 
the  perimeters  of  those  polygons. 


CIRCIES    SIMILAR.  165 

4T7.  By  the  method  of  iniinites,  the  circle  is  consid- 
ered as  a  regular  polygon  of  an  infinite  number  of 
sides,  each  side  being  an  infinitesimal  straight  line. 
But  the  method  of  limits  is  preferred  in  this  place,  be- 
cause, strictly  speaking,  the  circle  is  not  a  polygon,  and 
the  circumference  is  not  a  broken  line. 

The  above  theorem  establishes  only  this,  that  whatever 
is  true  of  all  inscribed,  or  of  all  circumscribed  polygons, 
is  necessarily  true  of  the  circle. 

478.  Theorem. — A  curve  is  shorter  than  any  other  line 
which  joins  its  ends,  and  toward  which  it  is  convex. 

For  the  curve  BDC  is  the 
limit  of  those  broken  lines 
which  have  their  vertices  in  it. 
Then,  the  curve  BDC  is  less 
than  the  line  BFC  (79). 

479.  Corollary — The  circumference  of  a  circle  is 
shorter  than  the  perimeter  of  a  circumscribed  polygon. 

480.  Corollary. — The  circumference  of  a  circle  is 
longer  than  the  perimeter  of  an  inscribed  polygon. 

This  is  a  corollary  of  the  Axiom  of  Distance  (54). 

481.  Theorem — A  circle  has  a  less  perimeter  than  any 
equivalent  polygoyi. 

For,  of  equivalent  polygons,  that  has  the  least  perim- 
eter which  is  regular  (468),  and  has  the  greatest  number 
of  sides  (470). 

482.  Corollary. — A  circle  has  a  greater  area  than 
any  isoperimetrical  figure. 

CIRCLES    SIMILAR. 
48S.  Theorem — Circles  are  similar  figures.  1 

For  angles  which  intercept  like  parts  of  a  circumfer- 
ence   are  equal  (207  and  224).     Hence,  whatever  lines 


i6()  ELEMENTS    OF    GEOMETRY. 

be  made  in  one  circle,  homologous  lines,  making  cqu^il 
angles,  may  be  made  in  another. 

This  theorem  may  be  otherwise  demonstrated,  thus: 
Inscribed  regular  polygons  of  the  same  number  of  sides 
are  similar.  The  number  of  sides  may  be  increased 
indefinitely,  and  the  polygons  will  still  be  similar  at  eacli 
successive  step.  The  circles  being  the  limits  of  the 
polygons,  must  also  be  similar. 

484.  Theorem. — Ttvo  sedors  are  similar  when  tlie  an- 
gles made  hy  their  radii  are  equal. 

485.  Theorem — Two  segments  are  similar  when  the 
angles  which  are  formed  hy  radii  from  the  ends  of  their 
respective  arcs  are  equal. 

These  two  theorems  are  demonstrated  by  completing 
the  circles  of  which  the  given  figures  form  parts.  Then 
the  given  straight  lines  in  one  circle  are  homologous  to 
those  in  the  other:  and  any  angle  in  one  may  have  its 
corresponding  equal  angle  in  the  other,  since  the  circles 
are  similar. 

EXERCISE. 

486.  When  the  Tyrian  Princess  stretched  the  thongs  cut  from 
the  hide  of  a  bull  around  the  site  of  Carthage,  what  course  should 
she  have  pursued  in  order  to  include  the  greatest  extent  of  terri- 
tory ? 

RECTIFICATION    OF    CIRCUMFERENCE. 

487.  Theorem. —  The  ratio  of  the  circumference  to  its 
diameter  is  a  constant  qiiantity. 

Two  circumferences  are  to  each  other  in  the  ratio  of 
their  diameters.  For  the  perimeters  of  similar  regular 
polygons  are  in  the  ratio  of  homologous  lines  (435); 
and  the  circumference  is  the  limit  of  the  perimeters  of 


RECTIFICATION    OF    CIRCUMFERENCE.  167 

regular  polygons  (476).  Then,  designating  any  two 
circumferences  by  C  and  C,  and  their  diameters  by  D 
and  W, 

C  :  C'  :  :  D  :  D'. 

Hence,  by  alternation, 

C  :  D  :  :  C  :  D^ 

That  is,  the  ratio  of  a  circumference  to  its  diametei 
is  the  same  as  that  of  any  other  circumference  to  its 
diameter. 

488.  The  ratio  of  the  circumference  to  the  diameter 
is  usually  designated  by  the  Greek  letter  ;r,  the  initial 
of 'perimeter. 

If  we  can  determine  this  numerical  ratio,  multiplying 
any  diameter  by  it  will  give  the  circumference,  or  a 
straight  line  of  the  same  extent  as  the  circumference. 
This  is  called  the  rectification  of  that  curve. 

489.  The  number  t:  is  less  than  4  and  greater  than 
3.  For,  if  the  diameter  is  1,  the  perimeter  of  the  cir- 
cumscribed square  is  4;  but  this  is  gi*eater  than  the 
circumference  (479).  And  the  perimeter  of  the  in- 
scribed regular  hexagon  is  3,  but  this  is  less  than  the 
circumference  (480). 

In  order  to  calculate  this  number  more  accurately,  let 
Us  first  establish  these  two  principles  : 

400.  Theorem. — Given  the  apothem,  radius,  and  side 
of  a  regular  polygon ;  the  apothem  of  a  regular  polygon  of 
the  same  length  of  perimeter,  hut  double  the  number  of 
sides,  is  half  the  sum  of  the  given  apothem  and  radius; 
and  the  radius  of  the  polygon  of  double  the  number  of 
sides,  is  a  mean  proportional  between  its  own  apothem  and 
the  given  radius. 

Let  CD  be  the  apothem,  CB  the  radius,  and  BE  the 
side  of  a  regular  polygon.     Produce  DC  to  F,  making 


168 


ELEMENTS    OF   GEOMETRY 


OF  equal  to  CB.     Join  BF  and  EF.     From  C  let  the 
perpendicular    CG    fall    upon    BF. 
Make  GH  parallel  to  BE,  and  join 
CH  and  CE. 

Now,  the  triangle  BCF  being  isos- 
celes by  construction,  the  angles  CBF 
and  CFB  are  equal.  The  sum  of 
these  two  is  equal  to  the  exterior  an- 
gle BCD  (261).  Hence,  the  angle 
BFD  is  half  the  angle  BCD.     Since  \  \    i 

DF  is,  by  hypothesis,  perpendicular  \  |  / 

to  BE  at  its  center,  BCE  and  BFE  \j/ 

are  isosceles  triangles  (108),  and  the  |, 

angles  BCE  and  BFE  are  bisected 
by   the   line   DF  (271).      Therefore,  the  angle   BFE   is 
half  the  angle  BCE.     That  is,  the  angle  BFE  is  equal  to 
the  angle  at  the  center  of  a  regular  polygon  of  double 
the  number  of  sides  of  the  given  polygon  (447). 

Since  GH  is  parallel  to  BE, 

We  have,  GH  :  BE  : :  GF  :  BF. 

Since  GF  is  the  half  of  BF  (271),  GH  is  the  half  of 
BE.  Then  GH  is  equal  to  the  side  of  a  regular  poly- 
gon, with  the  same  length  of  perimeter  as  the  given 
polygon,  and  double  the  number  of  sides. 

Again,  FH  and  FG,  being  halves  of  equals,  are  equal. 
Also,  IF  is  perpendicular  to  GH  (127).  Therefore,  we 
have  GH  the  side,  IF  the  apothem,  and  GF  the  radius 
of  the  polygon  of  double  the  number  of  sides,  with  a 
perimeter  equal  to  that  of  the  given  polygon. 

Now,  the  similar  triangles  give, 

FI  :  FD  :  :  FG  :  FB. 

Therefore,  FI  is  one-half  of  FD.  But  FD  is,  by  con- 
struction, equal  to  the  sum  of  CD  and  CB.     Therefore, 


RECTIFICATION    OF    CIRCUMFERENCE.  l(j<j 

the  apothem  of  the  second  polygon  is  equal  to  half  the 
sum  of  the  given  apothem  and  radius. 

Again,  in  the  right  angled  triangle  GCF  (324), 

FC  :  FG  :  :  FG  :  FL 

But  FC  is  equal  to  CB;  therefore,  FG,  the  radius  of 
the  second  polygon,  is  a  mean  proportional  between  the 
given  radius  and  the  apothem  of  the  second. 

491-  For  convenient  application  of  these  principles, 
let  us  represent  the  given  apothem  by  a,  the  radius  by 
r,  and  the  side  by  s,  the  apothem  of  the  polygon  of 
double  the  number  of  sides  by  x,  and  its  radius  by  y. 

Then,  x  =  — ^,  and  x  :  y  \  :  y  :  r. 

Hence,  i/^  =  a:r,  and  y  =  \/ xr. 

493.  Again,  since,  in  any  regular  polygon,  the  apo- 
them, radius,  and  half  the  side  form  a  right  angled 
triangle, 

We  always  have,         r^  =  «^-f- 1  9  I 


Hence,  a=^y\r'^ — 7  =  2l/4r^ 


49S.  Problem. — To  find  the  approximate  value  of  the 
ratio  of  the  circumference  to  the  diameter  of  a  circle. 

Suppose  a  regular  hexagon  whose  perimeter  is  unity. 
Then  its  side  is  J  or  .166667,  and  its  radius  is  the 
same  (454). 


By  the  formula,    a  =  J|/4r'^ — s^,  the  apothem  is 
iV7^=^\  =  i'-2V'%  or  .144338. 

Then,  by  the   formula,  x  =  i  (a-{-r),  the   apothem  of 
the  regular  polygon  of  twelve  sides,  the  perimeter  being 

unity,  is  i(i  +  riV^)  or  .155502.     The  radius  of  the 
Geom. — 15 


170  ELEMENTS    OF    GEOMETRY. 

same,  by  the   formula  i/=^y^xr,  is   .160988.     Proceed 
ing  in  the  same  way,  the  following  table  may  be  con- 
structed : 
I 

REGULAR  POLYGONS  WHOSE  PERIMETER 


IS  UNITY. 

Kumber  of  sides. 

Apothem. 

Radius. 

6 

.144338 

.166667 

12 

.155502 

.160988 

24 

.158245 

.159610 

48 

.158928 

.159269 

96 

.159098 

.159183 

192 

.159141 

.159162 

384 

.159151 

.159157 

768 

.159154 

.159155 

1536 

.159155 

.159155 

Now,  observe  that  the  numbers  in  the  second  column 
express  the  ratios  of  the  radius  of  any  circle  to  the 
perimeters  of  the  circumscribed  regular  polygons;  and 
that  those  in  the  third  column  express  the  ratios  of  the 
radius  to  the  perimeters  of  the  inscribed  polygons. 
These  ratios  gradually  approach  each  other,  till  they 
agree  for  six  places  of  decimals.  It  is  evident  that  by 
continuing  the  table,  and  calculating  the  ratios  to  a 
greater  number  of  decimal  places,  this  approximation 
could  be  made  as  near  as  we  choose. 

But  it  has  been  already  shown  that  the  circumference 

is  less   than   the   perimeter   of  the   circumscribed,   and 

greater  than  that  of  the  inscribed  polygon.     Hence,  we 

conclude,  that  when  the  circumference  is  1,  the  radius 

is  .159155,  with    a    near   approximation   to    exactness. 

The  diameter,  being  double  the  radius,  is  .31831. 

Therefore, 

T=^ThT  =  3-14159. 


RECTIFICATION    OF    CIRCUMFERENCE.  171 

494.  It  was  shown  by  Archimedes,  by  methods  resem- 
bling the  above,  that  the  value  of  ;:  is  less  than  3^,  and 
greater  than  3  J  J.  This  number,  3],  is  in  very  common 
use  for  mechanical  purposes.  It  is  too  great  by  about 
one  eight-hundredth  of  the  diameter. 

About  the  year  1640,  Adrian  Metius  found  the  nearer 
approximation  ff |,  which  is  true  for  six  places  of  deci- 
mals. It  is  easily  retained  in  the  memory,  as  it  is  com- 
posed of  the  first  three  odd  numbers,  in  pairs,  113  355, 
taking  the  first  three  digits  for  the  denominator,  and  the 
other  three  for  the  numerator. 

By  the  integral  calculus,  it  has  been  found  that  ;:  is 
equal  to  the  series  4  —  ^  -f- 1  —  ^  +  y  —  A  +,  etc. 

By  the  calculus  also,  other  and  shorter  methods  have 
been  discovered  for  finding  the  approximate  value  of  tt. 
In  1853,  Mr.  Rutherford  presented  to  the  Royal  Society 
of  London  a  calculation  of  the  value  of  tt  to  five  hund- 
red and  thirty  decimals,  made  by  Mr.  W.  Shanks,  of 
Houghton-le-Spring. 

The  first  thirty-nine  decimals  are, 

3.141  592  653  589  793  238  462  643  383  279  502  884  197. 

EXERCISES. 

495. — 1.  Two  wheels,  whose  diameters  are  twelve  and  eighteen 
inches,  are  connected  by  a  belt,  so  that  tlie  rotation  of  one  causes 
that  of  tlie  other.  Tlie  smaller  makes  twenty-four  rotations  in  a 
minute;  what  is  the  velocity  of  the  larger  wheel? 

2.  Two  wheels,  whose  diameters  are  twelve  and  eightei^n  inches, 
are  fixed  on  the  same  axle,  so  that  they  turn  together.  A  point 
on  the  rim  of  the  smaller  moves  at  the  rate  of  six  feet  per  second; 
what  is  the  velocity  of  a  point  on  the  rim  of  the  larger  wheel  ? 

3.  If  the  radius  of  a  car-wlieel  is  thirteen  inches,  how  many 
revolutions  does  it  make  in  traveling  one  mile? 

4.  If  the  equatorial  diameter  of  the  earth  is  7924  miles,  what 
is  the  length  of  one  degree  of  longitude  on  the  equator? 


372 


ELEMEiNTS   OF   GEOMETRY. 


QUADRATURE    OF    CIRCLE. 

406.  -The  quadrature  oy -squaring  of  the  circle^  that  is, 
the  finding  an  equivalent  rectilinear  figure,  is  a  problem 
which  excited  the  attention  of  mathematicians  during 
many  ages,  until  it  was  demonstrated  that  it  could  only 
be  solved  approximately. 

The  solution  depends,  indeed,  on  the  rectification  of 
the  circumference,  and  upon  the  following 

497.  Theorem — The  area  of  any  polygon  in  which  a 
circle  can  he  inscribed,  is  measured  hy  half  the  product  of 
its  perimeter  by  the  radius  of  the  inscribed  circle. 

From   ;;he   center  C  of  the   circle,  let   straight   lines 
extend  to  all  the  vertices  of 
the  polygon  ABDEF,  also  to 
all  the  points  of  tangency,  G, 
H,  I,  K,  and  L. 

The  lines  extending  to  the 
points  of  tangency  are  radii  of 
the  circle,  and  are  therefore 
perpendicular  to  the  sides  of 
the  polygon,  which  are  tan- 
gents of  the  circle  (183).  The 
polygon  is  divided  by  the  lines 
extending  to  the  vertices  into 
as  many  triangles  as  it  has 
sides,  ACB,  BCD,  etc.  Re- 
garding, the  sides  of  the  poly- 
gon, AB,  BD,  etc.,  as  the  bases  of  these  several  trian- 
gles, they  all  have  equal  altitudes,  for  the  radii  are 
perpendicular  to  the  sides  of  the  polygon.  Now,  the 
area  of  each  triangle  is  measured  by  half  the  product 
of  its  base  by  the  common  altitude.  But  the  area  of 
the   polygon  is  the  sum  of  the  areas  of  the  triangles, 


QUADRATURE    OF    CIRCLE.  173 

and  the  perimeter  of  the  polygon  is  the  sum  of  their 
bases.  It  follows  that  the  area  of  the  polygon  is  meas- 
ured by  half  the  product  of  the  perimeter  by  the  com- 
mon altitude,  which  is  the  radius. 

498.  Corollary — The  area  of  a  regular  polygon  b. 
measured  by  half  the  product  of  its  perimeter  by  its 
apothem. 

499.  Theorem — The  area  of  a  circle  is  measured  hy 
half  the  product  of  its  circumference  hy  its  radius. 

For  the  circle  is  the  limit  of  all  the  polygons  that 
may  be  circumscribed  about  it,  and  its  circumference  is 
the  limit  of  their  perimeters. 

500.  Theorem. —  The  area  of  a  circle  is  equal  to  the 
square  of  its  radius,  multiplied  by  the  ratio  of  the  circum- 
ference to  the  diameter. 

For,  let  r  represent  the  radius.  Then,  the  diameter 
is  2  r,  and  the  circumference  is  ;r  X  2  r,  and  the  area  is 
J;rX2rXr,  or  7rr-  (499);  that  is,  the  square  of  the 
radius  multiplied  by  the  ratio  of  the  circumference  to 
the  diameter. 

501.  Corollary — The  areas  of  two  circles  are  to  each 
other  as  the  squares  of  their  radii ;  or,  as  the  squares  of 
their  diameters. 

502.  Corollary. — When  the  radius  is  unity,  the  area 

is  expressed  by  t:. 

503.  Theorem. — The  area  of  a  sector  is  measured  hy 
half  the  product  of  its  arc  hy  its  radius. 

For,  the  sector  is  to  the  circle  as  its  arc  is  to  the 
circumference.  This  may  be  proved  in  the  same  man- 
ner as  the  proportionality  of  arcs  and  angles  at  the  cen- 
ter (107  or  2C2). 

504.  Since  that  which  is  true  of  every  polygon  may 


174  ELEMENTS  OF  GEOMETRY. 

be  shown,  by  the  method  of  limits,  to  be  true  also  of 
plane  figures  bounded  by  curves,  it  follows  that  in  any 
two  similar  plane  surfaces  the  ratio  of  the  areas  is  the 
second  power  of  the  linear  ratio. 

505.  Some  of  the  following  exercises  are  only  arith- 
metical applications  of  geometrical  principles. 

The  algebraic  method  may  be  used  to  great  advantage 
in  many  exercises,  but  every  principle  or  solution  that 
is  found  in  this  Avay,  should  also  be  demonstrated  by 
geometrical  reasoning. 

EXERCISES. 

m 

5(^6. — 1.  What  is  the  length  of  the  radius  when  the  arc  of 
80°  is  10  feet  ? 

2.  What  is  the  value,  in  degrees,  of  the  angle  at  the  center, 
whose  arc  has  tlie  same  length  as  the  radius? 

3.  What  is  the  area  of  the  segment,  whose  arc  is  60°,  and  ra- 
dius 1  foot? 

4.  To  divide  a  circle  into  two  or  more  equivalent  parts  by  con- 
centric circumferences. 

5.  One-tenth  of  a  circular  field,  of  one  acre,  is  in  a  walk  ex- 
tending round  the  whole;  required  the  width  of  the  walk. 

6.  Two  irregular  garden-plats,  of  the  same  shape,  contain,  re- 
spectively, 18  and  32  square  yards;  required  their  linear  ratio. 

7.  To  describe  a  circle  equivalent  to  two  given  circles. 

50T.  The  following  exercises  may  require  the  student  to  re- 
view the  leading  principles  of  Plane  Geometry. 

1.  From  two  points,  one  on  each  side  of  a  given  straight  line, 
to  draw  lines  making  an  angle  that  is  bisected  by  the  given  line. 

2.  If  two  straight  lines  are  not  parallel,  the  difference  between 
the  alternate  angles  formed  by  any  secant,  is  constant. 

3.  To  draw  the  minimum  tangent  from  a  given  straight  line 
to  a  given  circumference. 

4.  How  many  circles  can  be  made  tangent  to  three  given 
straiglit  lines  ?  » 


EXERCISES.  175 

5.  Of  all   triangles  on  the  same  base,  and   having   the  same 
vertical  angle,  the  isosceles  has  the  greatest  area. 

6.  To  describe  a   circumference  through    a   given    point,  and 
touching  a  given  line  at  a  given  point. 

7.  To  describe  a  circumference  through  two  given  points,  and 
touching  a  given  straight  line. 

8.  To   describe   a  circumference  through   a  given    point,  and 
touching  two  given  straight  lines. 

9.  About  a  given  circle  to  describe  a  triangle  similar  to  a 
given  triangle. 

10.  To  draw  lines  having  the  ratios  ^/2,  |/3,  i/5,  etc. 

11.  To  construct  a  triangle  with  angles  in  the  ratio  1,  2,  3. 

12.  Can  two  unequal  triangles  have  a  side  and  two  angles  in 
the  one  equal  to  a  side  and  two  angles  in  the  other? 

13.  To  construct  a  triangle  when  the  three  lines  extending  from 
the  vertices  to  the  centers  of  the  opposite  sides  are  given? 

14.  If  two  circles  touch  each  other,  any  two  straight  lines  ex- 
tending through  the  point  of  contact  will  be  cut  proportionally 
by  the  circumferences. 

15  If  any  pomt  on  the  circumference  of  a  circle  circumscrib- 
ing an  equilateral  triangle,  be  joined  by  straight  lines  to  the  sev- 
eral vertices,  the  middle  one  of  these  lines  is  equivalent  to  the 
other  Lwo. 

16.  Making  two  diagonals  in  any  quadrilateral,  ihe  triangles 
formed  by  one  have  their  areas  in  the  ratio  of  the  parts  of  ihe  other. 

17.  To  bisect  any  quadrilateral  by  a  line  from  a  guven  vertex. 

18.  In  the  triangle  ABC,  the  side  AB=13,  BC  =^  15,  the  alti- 
tude=12;  requireu  the  base  AC. 

19.  The  sides  of  a  triangle  have  the  ratio  of  65,  70.  and  75; 
its  area  is  21  square  inches;   required  the  length  of  each  side. 

20.  To  inscribe  a  square  in  a  given  segment  of  a  circle. 

21.  If  any  point  within  a  parallelogram  be  joined  to  each  of 
the  four  vertices,  two  opposite  triangles,  thus  formed,  are  together 
equivalent  to  half  the  parallelogram. 

22.  To  divide  a  straight  line  into  two  such  parts  that  the  rect- 
angle contained  by  them  shall  be  a  maximum. 

23.  The  area  of  a  triangle  which  has  one  angle  of  30°,  is 
one-fourth  the  product  of  the  two  sides  containing  that  angle 


176  ELEMENTS  OF  GEOMETRY. 

24.  To  construct  a  riglit  angled  triangle  when  the  area  and 
hypotenuse  are  given. 

25.  Draw  a  right  angle  by  means  of  Article  413. 

26.  To  describe  four  equal  circles,  touching  each  other  exteri- 
orly, and  all  touching  a  given  circumference  interiorly. 

27.  A  chord  is  8  inches,  and  the  altitude  of  its  segment  3 
inches;    required  the  area  of  the  circle. 

28.  What  is  the  area  of  the  segment  whose  arc  is  3G°,  and 
chord  6  inches  ? 

29.  The  lines  which  bisect  the  angles  formed  by  producing  the 
sides  of  an  inscribed  quadrilateral,  are  perpendicular  to  each  other. 

30.  If  a  circle  be  described  about  any  triangle  ABC,  then, 
taking  BC  as  a  base,  the  side  AC  is  to  the  altitude  of  the  trian- 
gle as  the  diameter  of  the  circle  is  to  the  side  AB. 

31.  By  the  proportion  just  stated,  show  that  the  area  of  a  tri- 
angle is  measured  by  the  product  of  the  three  sides  multiplied 
together,  divided  by  four  times  the  radius  of  the  circumscribing 
circle. 

32.  In  a  quadrilateral  inscribed  in  a  circle,  the  sum  of  the  two 
rectangles  contained  by  opposite  sides,  is  equivalent  to  the  rect- 
an^e  contained  by  the  diagonals.  This  is  known  as  the  Ptolemaic 
Theorem. 

33.  Twice  the  square  of  the  straight  line  which  joins  the  vertex 
of  a  triangle  to  the  center  of  the  base,  added  to  twice  the  square 
of  half  the  base,  is  equivalent  to  the  sum  of  the  squares  of  the 
other  two  sides. 

34.  The  sum  of  the  squares  of  the  sides  of  any  quadrilateral  is 
equivalent  to  the  sum  of  the  squares  of  the  diagonals,  increased 
by  four  times  the  square  of  the  line  joining  the  centers  of  the 
diagonals. 

35.  If,  from  any  point  in  a  circumference,  perpendiculars  be  let 
fall  on  the  sides  of  an  inscribed  triangle,  the  three  points  of  inter- 
section will  be  in  the  same  straight  line. 


LINES    IN    SPACE. 


GEOMETRY   OF   SPACE. 

CHAPTER    IX. 
STRAIGHT    LINES    AND    PLANES. 

508.  The  elementary  principles  of  those  geometrical 
figures  which  lie  in  one  plane,  furnish  a  basis  for  the 
investigation  of  the  properties  of  those  figures  which  do 
not  lie  altogether  in  one  plane. 

We  will  first  examine  those  straight  figures  which  do 
not  inclose  a  space;  after  these,  certain  solids,  or  inclosed 
portions  of  space. 

The  student  should  bear  in  mind  that  when  straight 
lines  and  planes  are  given  by  position  merely,  without 
mentioning  their  extent,  it  is  understood  that  the  extent 
is  unlimited. 

LINES    IN    SPACE. 

SOO.  Theorem. —  Through  a  given  point  in  space  there 
can  he  only  one  line  parallel  to  a  given  straight  liiie. 

This  theorem  depends  upon  Articles  49  and  117,  and 
includes  Article  119. 

510.  Theorem. —  Tivo  straight  lines  in  space  parallel  to 
a  third,  are  parallel  to  each  other. 

This  is  an  immediate  consequence  of  the  definition  of 
parallel  lines,  and  includes  Article  118. 


173         .  ELEMENTS  OF  GEOMETRY. 

511«  Problem — There  may  he  in  space  any  number  of 
straight  lines,  each  perpendicular  to  a  given  straight  line 
at  one  point  of  it. 

For  we  may  suppose  that  while  one  of  two  perpen- 
dicular lines  remains  fixed  as  an  axis,  the  other  revolves 
around  it,  remaining  all  the  while  perpendicular  (48). 
Ta9  second  line  can  thus  take  any  number  of  positions. 

This  does  not  conflict  with  Article  103,  for,  in  this 
ca^e,  the  axis  is  not  in  the  same  plane  with  any  two  of 
the  perpendiculars. 

EXERCISES. 

512, — 1.  Designate  two  lines  which  are  everywhere  equally 
distant,  but  which  are  not  parallel. 

2.  Designate  two  straight  lines  which  are  not  ])arallel,  and  yet 
can  not  meet. 

3.  Designate  four  points  which  do  not  lie  all  in  one  plane. 

PLANE    AND    LINES. 

51S.  Theorem — The  position  of  a  plane  is  determined 
by  any  plane  figure  except  a  straight  line. 

This  is  a  corollary  of  Article  60. 

Hence,  we  say,  the  plane  of  an  angle,  of  a  circum- 
ference, etc. 

514.  Theorem — A  straight  line  arid  a  jt>Zawe  can  have 
only  one  common  point,  unless  the  line  lies  wholly  in  the  plane. 

This  is  a  corollary  of  Article  58. 

»I15^  When  a  line  and  a  plane  have  only  one  common 
point,  the  line  is  said  to  pierce  the  plane,  and  the  plane 
CO  cut  the  line.  The  common  point  is  called  the  foot  of 
the  line  in  the  plane. 

When  a  line  lies  wholly  in  a  plane,  the  plane  is  said 
to  pass  through  the  line. 


PLANE    AND    LINES, 


179 


516.  Theorem. — The  iyiter section  of  two  planes  is  a 
straight  line. 

For  two  planes  can  not  have  three  points  common, 
unless  those  points  are  all  in  one  straight  line  (59). 


PEEPENDICULAR    LINES. 

517.  Theorem. — A  straight  line  which  is  perpendicular 

to  each  of  two  straight  lines  at  their  point  of  intersection, 
is  perpendicular  to  every  other  straight  line  tvhich  lies  in 
the  plane  of  the  two,  and  passes  through  their  point  of 
intersection. 

In  the  diagram,  suppose  D,  B,  and  C   to  be  on  the 
plane  of  the  paper,  the  point  A 
being    above,  and    I    below    that 
plane. 

If  the  line  AB  is  perpendicu- 
lar to  BC  and  to  BD,  it  is  also 
perpendicular  to  every  other  line 
lying  in  the  plane  of  DBC,  and 
passing  through  the  point  B ;  as, 
for  example,  BE. 

Produce  AB,  making  BI  equal 
to  BA,  and  let  any  line,  as  FII, 

cut  the  lines  BC,  BE,  and  BD,  in  F,  G,  and  H.  Then 
join  AF,  AG,  AH,  and  IF,  IG,  and  IH. 

Now,  since  BC  and  BD  arc  perpendicular  to  AI  at  its 
center,  the  triangles  AFII  and  IFH  have  AF  equal  to 
IF  (108),  All  equal  to  III,  and  FH  common.  There- 
fore, they  are  equal,  and  the  angle  AHF  is  equal  to  IHF. 
Then  the  triangles  AHG  and  IHG  are  equal  (284),  and 
the  lines  AG  and  IG  are  equal.  Therefore,  the  line 
BG,  having  two  points  each  equally  distant  from  A  and 
I,  is  perpendicular  to  the  line  AI  at  its  center  B  (109). 


ISO 


ELEMENTS    OF    GEOMETRY 


In  the  same  way,  prove  that  any  other  line  through  B, 
in  the  plane  of  DBC,  is  perpendicular  to  AB. 

518.  Theorem Conversely,  if  several  straight  lines  are 

each  perpendicular  to  a  given  line  at  the  same  point,  then 
these  several  luus  all  lie  in  one  plane. 

Thus,  if  BA  is  perpendicular  to  BC,  to  BD,  and  tc 
BE,  then  these  three  all  lie  in  one  plane. 

BD,  for  instance,  must  be  in  the 
plane  CBE.  For  the  intersection 
of  the  plane  of  ABD  with  the  plane 
of  CBE  is  a  straight  line  (516). 
This  straight  intersection  is  per- 
pendicular to  AB  at  the  point  B 
(517).  Therefore,  it  coincides  with 
BD  (103).  Thus  it  may  be  shown 
that  any  other  line,  perpendicular 
to  AB  at  the  point  B,  is  in  the 
pl?.ne  of  C,  B,  D,  and  E. 

510.  A  straight  line  is  said  to  be  perpendicular  to  a 
plane  when  it  is  perpendicular  to  every  straight  line 
which  passes  through  its  foot  in  that  plane,  and  the 
plane  is  said  to  be  perpendicular  to  the  line.  Every  line 
not  perpendicular  to  a  plane  which  cuts  it,  is  called 
oblique. 

520.  Corollary. — If  a  plane  cuts  a  line  perpendicu- 
larly at  the  middle  point  of  the  line,  then  every  point 
of  the  plane  is  equally  distant  from  the  two  ends  of  the 
line  (108). 

531.  Corollary. — If  one  of  two  perpendicular  lines 
revolves  about  the  other,  the  revolving  line  describes  a 
plane  which  is  perpendicular  to  the  axis. 

f%22.  Corollary Through  one  point  of  a  straight  line 

there  can  be  only  one  plane  perpendicular  to  that  line. 


PLANE    AND    LINES. 


181 


c 

M            \ 

\ 

D B ""E 

523.  Theorem. — Through  a  jjoint  out  of  a  plane  there 
can  he  only  one  straight  line  perpendicular  to  the  plane. 

For,  if  there  could  be  two  perpendiculars,  then  each 
would  be  perpendicular  to  the  line  in  the  plane  which 
joins  their  feet  (519).     But  this  is  impossible  (103). 

•524.  Theorem. —  Through  a  point  in  a  plane  there  con 
he  only  one  straight  line  perpendicular  to  the  plane. 

Let  BA  be  perpendicular  to  the  plane  MN  at  the  point 
B.     Then  any  other 
line,  BC  for  example, 
will  be  oblique  to  the 
plane  MN. 

For,  if  the  plane 
of  ABC  be  produced, 
its  intersection  with 
the  plane  MN  will 
be  a  straight  line. 

Let  DE  be  this  intersection.  Then  AB  is  perpen- 
dicular to  DE.  Hence,  BC,  being  in  the  plane  of  A,  D, 
and  E,  is  not  perpendicular  to  DE  (103).  Therefore,  it 
is  not  perpendicular  to  the  plane  MN  (519). 

525.  Corollary — The  direction  of  a  straight  line  in 
space  is  fixed  by  the  fact  that  it  is  perpendicular  to  a 
given  plane. 

The  directions  of  a  plane  are  fixed  by  the  fact  that  it 
is  perpendicular  to  a  given  line. 

526.  Corollary — All  straight  lines  which  are  perpen- 
dicular to  the  same  plane,  have  the  same  direction  ;  that 
is,  they  are  parallel  to  each  other. 

527.  Corollary. — If  one  of  two  parallel  lines  is  per- 
pendicular to  a  plane,  the  other  is  also. 

528.  The  Axis  of  a  circle  is  the  straight  line  perpen- 
dicular to  the  plane  of  the  circle  at  its  center. 


182 


ELEMENTS    OF   GEOMETRY. 


M 

V 

B 

OBLIQUE    LINES    AND    PLANES. 

529.  Theorem. — If  from,  a  point  without  a  plane,  a 
perpendicular  and  oblique  lines  be  extended  to  the  plane, 
then  two  oblique  lines  which  meet  the  pla^ie  at  equal  dis^ 
tances  from  the  foot  of  the  perpendicular,  are  equal. 

Let  AB  be  perpendicular, 
and  AC  and  AD  oblique  to 
the  plane  MN,  and  the  dis- 
tances BC  and  BD  equal. 

Then  the  triangles  ABC 
and  ABD  are  equal  (284), 
and  AC  is  equal  to  AD. 

530.  Corollary. — A  perpendicular  is  the  shortest  line 
from  a  point  to  a  plane.  Hence,  the  distance  from  a 
point  to  a  plane  is  measured  by  a  perpendicular  line. 

531.  Corollary. — All  points  of  the  circumference  of 
1  circle  are  equidistant  from  any  point  of  its  axis. 

S33.  If  from  all  points  of  a  line  perpendiculars  be  let 
fall  upon  a  plane,  the  line  thus  described  upon  the  plane 
is  the  projection  of  the  given  line  upon  the  given  plane. 

533.  Theorem. —  The  projection  of  a  straight  line  upon 
a  plane  is  a  straight  line. 

Let  AB  be  the  given  line,  and  MN  the  given  plane. 
Then,  from  the  points  A 
and  B,  let  the  perpen- 
diculars, AC  and  BD, 
fall  upon  the  plane  MN. 
Join  CD. 

AC  and  BD,  being  per- 
pendicular to  the  same 
plane,  are  parallel  (526), 
and  lie  in  cne  plane  (121),     Now,  every  perpendicular 


PLANE    AND    LINES. 


183 


to  MN  let  fall  from  a  point  of  AB,  must  be  parallel  to 
BD,  and  must  therefore  lie  in  the  plane  AD,  and  meet 
the  plane  MN  in  some  point  of  CD.  Hence,  the  straight 
line  CD  is  the  projection  of  the  straight  line  AB  on  the 
plane  MN. 

There  is  one  exception  to  this  proposition.  When  the 
given  line  is  perpendicular  to  the  plane,  its  projection 
is  a  point. 

5S4.  Corollary. — A  straight  line  and  its  projection 
on  a  plane,  both  lie  in  one  plane. 

535.  Theorem — The  angle  which  a  straight  line  makes 
with  its  projection  on  a  plane,  is  smaller  than  the  angle  it 
makes  with  any  other  line  in  the  plane. 

Let  AC  be  the  given  line,  and  BC  its  projection  on 
the  plane  MN.  Then 
the  angle  ACB  is  less 
than  the  angle  made 
by  AC  with  any  other 
line  in  the  plane,  as  CD. 

With  C  as  a  center 
and  BC  as  a  radius,  de- 
scribe a  circumference 
in  the  plane  MN,  cut- 
ting CD  at  D. 

Then  the  triangles  ACD  and  ACB  have  two  sides  of 
che  one  respectively  equal  to  two  sides  of  the  other. 
But  the  third  side  AD  is  longer  than  the  third  side  AB 
(530).  Therefore,  the  angle  ACD  is  greater  than  the 
single  ACB  (294). 

536.  Corollary — The  angle  ACE,  which  a  line  makes 
with  its  projection  produced,  is  larger  than  the  angle 
made  with  any  other  line  in  the  plane. 

»    537.  The  angle  which  a  line  makes  Avith  its  projec 


184  ELEMENTS    OF    GEOMETRY. 

tion  in  a  plane,  is  called  the  Angle  of  Inclination  of  the 
line  and  the  plane. 


PARALLEL  LINES  AND  PLANE. 

538.  Theorem. — If  a  straight  line  in  a  plane  is  paral- 
lel to  a  straight  line  not  in  the  plane,  then  the  second  line 
and  the  plane  can  not  have  a  common  point. 

For  if  any  line  is  parallel  to  a  given  line  in  a  plane, 
and  passes  through  any  point  of  the  plane,  it  will  lie 
wholly  in  the  plane  (121).  But,  by  hypothesis,  the  sec- 
ond line  does  not  lie  wholly  in  the  plane.  Therefore,  it 
can  not  pass  through  any  point  of  the  plane,  to  what- 
ever extent  the  two  may  be  produced. 

539.  Such  a  line  and  plane,  having  the  same  direc- 
tion, are  called  parallel. 

5i:0.  Corollary — If  one  of  two  parallel  lines  is  par- 
allel to  a  plane,  the  other  is  also. 

5irl.  Corollary — A  line  which  is  parallel  to  a  plane 
is  parallel  to  its  projection  on  that  plane. 

fills.  Corollary. — A  line  parallel  to  a  plane  is  every- 
where equally  distant  from  it. 

APPLICATIONS. 

543.  Three  points,  however  placed,  must  always  be  in  the 
same  plane.  It  is  on  this  principle  that  stability  is  more  readily 
obtained  by  three  supports  than  by  a  greater  number.  A  three- 
legged  stool  must  be  steady,  but  if  there  be  four  legs,  their  ends 
should  he  in  (jne  plane,  and  the  floor  should  be  level.  Many  sur- 
veying and  astronomical  instruments  are  made  with  three  legs. 

544.  The  use  of  lines  perpendicular  to  planes  is  very  frequent 
in  the  mechanic  arts.  A  ready  way  of  constructing  a  line  perpen- 
dicular to  a  plane  is  by  the  use  of  two  squares  (114).  Place  the 
angle  of  each  at  the  foot  of  the  desired  perpendicular,  one  side  o^ 


DlEDllAL    ANGLES.  185 

each  square  resting  on  the  plane  surface.  Bring  their  perpendic- 
ular sides  together.  Their  position  must  tlien  be  that  of  a  per- 
pendicular to  the  plane,  for  it  is  perp(!ndicular  to  two  lines  in  the 
plane. 

545.  When  a  circle  revolves  round  its  axis,  the  figure  under- 
goes no  real  change  of  position,  each  point  of  the  circumference 
taking  successively  the  position  deserted   by  another  point. 

On  this  principle  is  founded  the  operation  of  millstones.  Two 
circular  stones  are  placed  so  as  to  have  tlie  same  axis,  to  which 
their  faces  are  perpendicular,  being,  therefore,  parallel  to  each 
other.  The  lower  stone  is  fixed,  while  the  upper  one  is  made  to 
revolve.  The  relative  position  of  the  faces  of  the  stones  under- 
goes no  change  during  the  revolution,  and  their  distance  being 
properly  regulated,  all  the  grain  which  passes  between  them  will 
be  ground  with  the  same  degree  of  fineness. 

546.  In  the  turning  lathe,  the  axis  round  which  the  body  to 
be  turned  is  made  to  revolve,  is  the  axis  of  the  circles  formed  by 
the  cutting  tool,  which  removes  the  matter  projecting  beyond  a 
proper  distance  from  the  axis.  The  cross  section  of  every  part  of 
the  thing  turned  is  a  circle,  all  the  circles  having  the  same  axis. 


DIEDRAL    ANGLES. 

547.  A  DiEDRAL  Angle  is  formed  by  two  planes 
meeting  at  a  common  line.  This  figure  is  also  called 
simply  a  diedral.  The  planes  are  its  faces^  and  the  in- 
tersection is  its  edge. 

In  naming  a  diedral,  four  letters  are  used,  one  in  each 
fr.ce,  and  two  on  the  edge,  the  letters  on  the  edge  being 
between  the  other  tAvo. 

This  figure  is  called  a  diedral  angle^  because  it  is  simi- 
lar in  many  respects  to  an  angle  formed  by  two  lines. 

MEASURE    OF    DIEDRALS. 

548.  The  quantity  of  a  diedral,  as  is  the  case  with 
a  linear  angle,  depends  on  the  difference  in  the  directions 

Oconi. — If) 


186 


ELEMENTS   OF   GEOMETRY 


of  the  faces  from  the  edge,  without  regard  to  the  extent 
of  the  planes.  Hence,  two  diedrals  are  equal  when  they 
can  be  so  placed  that  their  planes  will  coincide. 

519.  Problem. — One  diedral  may  he  added  to  another. 

In  the  diagram,  AB,  AC,  and  AD 
represent  three  planes  having  the 
common  intersection  AE. 

Evidently  the  sum  of  BEAC  and 
OEAD  IS  equal  to  BEAD. 

550.  Corollary Diedrals     may 

be  subtracted  one  from  another.  A 
diedral  may  be  bisected  or  divided  in 
any  required  ratio  by  a  plane  pass- 
ing through  its  edge. 

551.  But  there  are  in  each  of  these  planes  any  num- 
ber of  directions.  Hence,  it  is  necessary  to  determine 
which  of  these  is  properly  the  direction  of  the  face  from 
the  edge.  For  this  purpose,  let  us  first  establish  the 
following  principle: 

552.  Theorem. — One  diedral  is  to  another  as  the  plane 
angle,  formed  in  the  first  hy  a  line  in  each  face  perpen- 
dicular to  the  edge,  is  to  the  similarly  formed  angle  in  the 
other. 

Thus,  if  FO,  GO,  and 
HO  are  each  perpendicu- 
lar to  AE,  then  the  die- 
dral CEAD  is  to  the  die- 
dral BEAD  as  the  angle 
GO  11  is  to  the  angle 
FOR.  This  may  be  de- 
monstrated in  the  same 
manner  as  the  proposi- 
tion in  Article  197. 


DIEDRAL   ANGLES. 


187 


553.  Corollary. — A  diedral  is  said  to  be  measured 
by  the  plane  angle  formed  by  a  line  in  each  of  its  faces 
perpendicular  to  the  edge. 

554.  Corollary. — Accordingly,  a  diedral  angle  may  be 
acute,  obtuse,  or  right.  In  the  last  case,  the  planes  are 
perpendicular  to  each  other. 

555.  Many  of  the  principles  of  plane  angles  may  be 
applied  to  diedrals,  without  further  demonstration. 

All  right  diedral  angles  are  equal  (90). 

When  the  sum  of  several  diedrals  is  measured  by 
two  right  angles,  the  outer  faces  form  one  plane  (100). 

When  two  planes  cut  each  other,  the  opposite  or  ver- 
tical diedrals  are  equal  (99). 


PERPENDICULAR    PLANES. 

556.  Theorem — If  a  line  is  perpendicular  to  a  plane^ 
then  any  plane  passing  through  this  line  is  perpendicular 
to  the  other  plane. 

If  AB  in  the  plane  PQ  is  perpendicular  to  the  plane 
MN,  then  AB  must  be  perpen- 
dicular to  every  line  in  MN 
which  passes  through  the 
point  B  (519);  that  is,  to  RQ, 
the  intersection  of  the  two 
planes,  and  to  BC,  which  is 
made  perpendicular  to  the  in- 
tersection RQ.  Then,  the  an- 
gle ABC  measures  the  inclina- 
tion of  the  two  planes  (553),  and  is  a  right  angle.  There- 
fore, the  planes  are  perpendicular. 

557.  Corollary. — Conversely,  if   a   plane   is   perpen- 
dicular to  another,  a  straight  line,  which  is  perpendicu- 


p 

^ 

V 

M 

R 

.^ 

c'"-"^^ 

Q 

N 


188  ELEMENTS  OF  GEOMETRY 

lar  to  one  of  them,  at  some  point  of  their  intersection, 
must  lie  wholly  in  the  other  phine  (524). 

*%^3,  Corollary — If  two  planes  are  perpendicular  to 
a  third,  then  the  intersection  of  the  first  two  is  a  line 
perpendicular  to  the  third  plane. 

OBLIQUE    PLANES. 

559.  Theorem — If  from  a  point  within  a  diedral,  per- 
pendicular lines  be  made  to  the  two  faces,  the  angle  of 
these  lines  is  supplementary  to  the  angle  which  measures 
the  diedral. 

Let  M  and  N  be  two  planes  whose  intersection  is 
AB,  and  CF  and  CE  perpendicu- 
lars let  fall  upon  them  from 
the  point  C;  and  DF  and  DE 
the  intersections  of  the  plane 
FOE  with  the  two  planes  M 
and  N.  Then  the  plane  FCE 
mist  be  perpendicular  to  each 
of  the   planes  M  and  N  (556). 

Hence,  the  line  AB  is  perpendicular  to  the  plane  FCE 
(558),  and  the  angles  ADF  and  ADE  are  right  angles. 
Then  the  angle  FDE  measures  the  diedral.  But  in  the 
quadrilateral  FDEC,  the  two  angles  F  and  E  are  right 
angles.  Therefore,  the  other  two  angles  at  C  and  D  are 
supplementary. 

560.  Theorem — Every  point  of  a  plane  which  bisects 
a  diedral  is  equally  distajit  from  its  two  faces. 

Let  the  plane  FC  bisect  the  diedral  DBCE.  Then  it 
is  to  be  proved  that  every  point  of  this  plane,  as  A,  for 
example,  is  equally  distant  from  the  planes  DC  and  EC. 

From  A  let  the  perpendiculars  AH  and  AI  fall  upon 
the  faces  DC  and  EC,  and  let  10,  AO,  and  IIO  be  the 


DIEDKAL    ANGLES.  189 

intersections  of  the  plane  of  the  angle  lAH  Avith  the 
three  given  planes. 

Then  it  may  be  shown,  as  in  the  last  theorem,  that  the 
angle  HOA  measures 

the  diedral  FBCD,  and  . 

the     angle    lOA    the  y^ 

diedral  FBCE.      But  z^^^^nT/^  / 

these      diedrals      are       d^  -/   ^<^  •  L 

equal,  by  hypothesis.  ^v  H^^   Ax^f  Z^ 

Therefore,  the  line  AG  x^^Axo'"'^        y^ 

bisects  the  angle  lOH,  B  E 

and  the  point  A  is  equally  distant  from  the  lines  OH 
and  01  (113).  But  the  distance  of  A  from  these  lines 
is  measured  by  the  same  perpendiculars,  AH  and  AI, 
which  measure  its  distance  from  the  two  faces  DC  and 
EC.  Therefore,  any  point  of  the  bisecting  plane  is 
equally  distant  from  the  two  faces  of  the  given  diedral. 

APPLICATIONS. 

561.  Articles  548  to  554  are  illustrated  by  a  door  turning  on 
its  hinges.  In  every  position  it  is  perpendicular  to  the  floor  and 
ceiling.  As  it  turns,  it  changes  its  inclination  to  the  wall,  in 
which  it  is  constructed,  the  angle  of  inclination  being  that  which 
is  formed  by  the  upper  edge  of   the  door  and  the  lintel. 

562.  The  theory  of  diedrals  is  as  important  in  the  study  of 
magnitudes  bounded  by  planes,  as  is  the  theory  of  angles  in  the 
study  of  polygons. 

This  is  most  striking  in  the  science  of  crystallography,  which 
teaches  us  how  to  classify  mineral  substances  according  to  their 
geometrical  forms.  Crystals  of  one  kind  have  edges  of  which  the 
diedral  angles  measure  a  certain  number  of  degrees,  and  crystals 
of  another  kind  have  edges  of  a  diflTerent  number  of  degrees. 
Crystals  of  many  species  may  be  thus  classified,  by  measuring 
their  diedrals. 

563.  The  i)laiie  of  the  surface  of  a  liquid  at  rest  is  called  hori- 
zontal, or  the   plane  of  the   horizon.     The  direction  of  a  phnnb- 


J  90  ELEMENTS   OF   GEOMETRY. 

line  when  the  weight  is  at  rest,  is  a  vertical  line.  The  vertical 
line  is  perpendicular  to  the  horizon,  the  positions  of  both  being 
governed  by  the  same  causes.  Every  line  in  the  plane  of  the 
horizon,  or  parallel  to  it,  is  called  a  horizontal  line,  and  every 
[)lane  passing  through  a  vertical  line  is  called  a  vertical  plane. 
Every  vertical  plane  is  perpendicular  to  the  horizon. 

Horizontal  and  vertical  planes  are  in  most  frequent  use.  Floors, 
ceilings,  etc.,  are  examples  of  the  former,  and  walls  of  the  latter. 
The  methods  of  using  the  builder's  level  and  plummet  to  determ- 
ine the  position  of  these,  are  among  the  simplest  applications  of* 
geometrical  principles. 

Civil  engineers  have  constantly  to  observe  and  calculate  the 
[>osition  of  horizontal  and  vertical  planes,  as  all  objects  are  re- 
ferred to  these.  The  astronomer  and  the  navigator,  at  every  step, 
refer  to  the  horizon,  or  to  a  vertical  plane. 


EXERCISES. 

564. — 1.  If,  from  a  point  without  a  plane,  several  equal  oblique 
lines  extend  to  it,  they  make  equal  angles  with  the  plane. 

2.  If  a  line  is  perpendicular  to  a  plane,  and  if  from  its  foot  a 
perpendicular  be  let  fall  on  some  other  line  which  lies  in  the  plane, 
then  this  last  line  is  perpendicular  to  the  plane  of  the  other  two. 

3.  What  is  the  locus  of  those  points  in  space,  each  of  which 
is  equally  distant  from  two  given  points? 


PARALLEL    PLANES. 

565.  Two  planes  which  are  perpendicular  to  the 
same  straight  line,  at  different 
points  of  it,  are  both  fixed  in  po- 
sition (525),  and  they  have  the 
same  directions.  If  the  parallel 
lines  AB  and  CD  revolve  about 
the  line  EF,  to  which  they  are 
both  perpendicular,  then  each  of  the 
revolving  lines  describes   a   plane. 


Every  direction    assumed  by  one   line   is   the   same   ns 


PARALLEL    PLANES.  191 

that  of  the  other,  and,  in  the  course  of  a  complete  revo- 
lution, they  take  all  the  possible  directions  of  the  two 
planes. 

Two  planes  which  have  the  same  directions  are  called 
jKiraUel  planes. 

Parallelism  consists  in  having  the  same  direction, 
whether  it  be  of  two  lines,  of  two  planes,  or  of  a  line 
and  a  plane. 

560.  Corollary. — Two  planes  parallel  to  a  third  are 
parallel  to  each  other. 

567.  Corollary.  —  Two  planes  perpendicular  to  the 
same  straight  line  are  parallel  to  each  other. 

568.  Corollary. — A  straight  line  perpendicular  to  one 
of  two  parallel  planes  is  perpendicular  to  the  other. 

569.  Corollary. — Every  straight  line  in  one  of  two 
parallel  planes  has  its  parallel  line  in  the  other  plane. 
Therefore,  every  straight  line  in  one  of  the  planes  is 
parallel  to  the  other  plane. 

570.  Corollary. — Since  through  any  point  in  a  plane 
there  may  be  a  line  parallel  to  any  line  in  the  same 
plane  (121),  therefore,  in  one  of  two  parallel  planes, 
and  through  any  point  of  it,  there  may  be  a  straight 
line  parallel  to  any  straight  line  in  the  other  plane. 

571.  Theorem. — Two  parallel  planes  can  not  meet. 
For,  if  they  had  a  common  point,  being  parallel,  they 

would  have  the  same  directions  from  that  point,  and 
therefore  would  coincide  throughout,  and  be  only  one 
plane. 

572.  Theorem — The  intersections  of  two  parallel  planes 
hy  a  third  plane  are  parallel  lifies. 

Let  AB  and  CD  be  the  intersections  of  the  two  par- 
allel planes  M  and  N,  with  the  plane  P. 

Now,  if  through  C  there  be  a  line  parallel  to  AB,  it 


192 


ELEMENTS  OF  GEOMETRY. 


M/ 

A     1 

B 

C    1 

N/ 

N    / 

must  lie  in  the  plane  P  (121),  and  also  in  the  plane  N 
(570).     Therefore,  it  is  the  in- 
tersection CD,  and  the  two  in- 
tersections are  parallel  lines. 

When  two  parallel  planes 
are  cut  by  a  third  plane,  eight 
diedrals  are  formed,  which  have 
properties  similar  to  those  of 
Articles  124  to  128. 

573.  Theorem —  The  parts  of  two  parallel  lines  inter- 
cepted between  parallel  planes  are  equal. 

For,  if  the  lines  AB  and  CD  are  parallel,  they  lie  in 
one  plane.  Then  AC  and  BD 
are  the  intersections  of  this 
plane  with  the  two  parallel 
planes  M  and  P.  Hence,  AC 
is  parallel  to  BD,  and  AD  is  a 
parallelogram.  Therefore,  AB 
is  equal  to  the  opposite  side  CD. 

574.  Theorem. —  Two  'parallel  planes  are  everywhere 
equally  distant. 

For  the  shortest  distance  from  any  point  of  one  plane 
to  the  other,  is  measured  by  a  perpendicular.  But 
these  perpendiculars  are  all  parallel  (526),  and  therefore 
equal  to  each  other. 

57<S.  Theorem — If  the  two  sides  of  an  angle  are  each 
parallel  to  a  given  plane,  then  the  plane  of  that  angle  is 
parallel  to  the  given  plane. 

If  AB  and  AC  are  each  parallel  to  the  plane  M, 
then  the  plane  of  BAC  is  parallel  to  the  plane  M. 

From  A  let  the  perpendicular  AD  fall  upon  the  plane 
M,  and  let  the  projections  of  AB  and  AC  on  the  plane 
M  be  respectively  DE  and  DF. 


M 

A 

C 

P 

B 

J) 

PARALLEL    PLANES. 


193 


Since  DE  is  parallel  to  AB  (541),  DA  is  perpendic- 
ular to  AB  (127).  For  a 
like  reason,  DA  is  per- 
pendicular to  AC.  There- 
fore, DA  is  perpendicular 
to  the  plane  of  BAC  (517), 
and  the  two  planes  being 
perpendicular  to  the  same 
line  are  parallel  to  each 
other  (567). 

576.  Theorem. — If  two  straight  lines  which  cut  each 
other  are  respectively  parallel  to  two  other  straight  lines 
zvhich  cut  each  other,  then  the  plane  of  the  first  two  is 
parallel  to  the  plaiie  of  the  second  two. 

Let  AB  be  parallel  to  EF,  and  CD  parallel   to   GH. 
Then  the  planes  M  and  P 
are  parallel. 

For  AB  being  parallel  ^><r'  M 

to  EF,  is  parallel  to  the 
plane  P  in  which  it  lies 
(538).  Also,  CD  is  par- 
allel to  the  plane  P,  "for 
the  same  reason.  There- 
fore, the  plane  M  is  par- 
allel to  the  plane  P  (575). 

577.  Corollary. — The  angles  made  by  the  first  two 
lines  are  respectively  the  same  as  those  made  by  the  sec- 
ond two.  For  they  are  the  differences  between  the  same 
directions. 

This  includes  the  corresponding  principle  of  Plane 
Geometry. 

578.  Theorem. — Straight  lines  cut  hy  three  parallel 
julanes  are  divided  proportionally. 

If  the  line  AB  is  cut  at  the  points  A,  E,  and  B,  and 
G^om. — 17 


^"^^\ 

^^D 

c--"^ 

^~^B 

^"^-^ 

^^^-H 

G-^"^ 

^^--F      , 

194 


ELEMENTS  OF  GEOMETRY. 


the  line  CD  at  the  points  C,  F,  and  D,  by  the  paralU-l 
planes  M,  N,  and  P,  then 
AE  :  EB  :  :  CF  :  FD. 

Join  AC,  AD,  and  BD. 
AD  pierces  the  plane  N  in 
the  point  G.  Join  EG  and 
GF. 

Now,  EG  and  BD  are  par- 
allel, being  the  intersections 
of  the  parallel  planes  N  and 
P  by  the  third  plane  ABD 
(572).     Hence  (313), 

AE  :  EB  :  : 

For  a  like  reason, 

AG  :  GD  :  :  CF  :  FD. 

Therefore,        AE  :  EB  : :  CF  :  FD. 


AG  :  GD. 


APPLICATION 


579.  The  general  problem  o^  perspective  in  drawing,  consists 
in  representing  upon  a  plane  surface'  the  apparent  form  of  ob- 
jects in  sight.  This  plane,  the  plane  of  the  picture,  is  supposed 
to  be  between  the  eye  and  the  objects  to  be  drawn.  Then  each 
object  is  to  be  represented  upon  the  plane,  at  the  point  where  it 
would  be  pierced  by  the  visual  ray  from  the  object  to  the  eye. 

All  the  visual  rays  from  one  straight  object,  such  as  the  top 
of  a  wall,  or  one  corner  of  a  house,  lie  in  one  plane  (60).  Their 
intersection  with  the  plane  of  the  picture  must  be  a  straight  line 
(516).  Therefore,  all  straight  objects,  whatever  their  position, 
must  be  drawn  as  straight  lines. 

Two  parallel  straight  objects,  if  they  are  also  parallel  to  the 
plane  of  the  picture,  will  remain  parallel  in  the  perspective.  For 
the  lines  drawn  must  be  parallel  to  the  objects  (572),  and  there- 
fore to  each  other. 

Two  parallel  lines,  which  are  not  parallel  to  the  plane  of  the 
picture,  will  meet  in  the  perspective.     They  will  meet,  if  procjuced. 


TRIEDIIALS.  195 

at  that  point  where  the  plane  of  the  picture  is  pierced  by  a  line 
from  the  eye  parallel  to  the  given  lines. 


EXERCISES. 

5SO. — 1.  A  straight  line  makes  equal  angles  with  two  paral- 
lel planes. 

2.  Two  parallel  lines  make  the  same  angle  of  inclination  with 
a  given  plane. 

3.  The  projections  of  two  parallel  lines  on  a  plane  are  parallel. 

4.  When  two  planes  are  each  perpendicular  to  a  third,  and  their 
intersections  with  the  third  plane  are  parallel  lines,  then  the  two 
planes  are  parallel  to  each  other. 

5.  If  two  straight  lines  be  not  in  the  same  plane,  one  straight 
line,  and  only  one,  may  be  perpencijcular  to  both  of  them. 

6.  Demonstrate  the  last  sentence  of  Article  579. 


TRTEDRALS. 

581.  When  three  planes  cut  each  other,  three  cases 
are  possible. 

1st.  The  intersections  may 
coincide.  Then  six  diedrals 
are  formed,  having  for  their 
common  edge  the  intersection 
of  the  three  planes. 


2d.  The  three  intersections 
may  be  parallel  lines.  Then 
one  plane  is  parallel  to  the 
intersection  of  the  other  two. 


196 


ELEMENTS   OF   GEOMETRY. 


3d.  The  three   intersections  may  meet  at  one  point. 
Then  the  space  about 
the   point  is  divided 
by  the   three   planes 
into  eight  parts. 

The  student  will 
apprehend  this  better 
when  he  reflects  that 
two  intersecting 
planes  make  four  di- 
edrals.  Now,  if  a 
third    plane     cut 

through  the  intersection  of  the  first  two,  it  will  divide 
each  of  the  diedrals  into  two  parts,  making  eight  in  all. 
Each  of  these  parts  is  called  a  triedral,  because  it  has 
three  faces. 

A  fourth  case  is  impossible.  For,  since  any  two  of 
the  intersections  lie  in  one  plane,  they  must  either  be 
parallel,  or  they  meet.  If  two  of  the  intersections  meet, 
the  point  of  meeting  must  be  common  to  the  three 
planes,  and  must  therefore  be  common  to  all  the  in- 
tersections. Hence,  the  three  intersections  either  have 
more  than  one  point  common,  only  one  point  common, 
or  no  point  common.  But  these  are  the  three  cases 
just  considered. 

5S2.  A  Triedral  is  the  figure  formed  by  three  planes 
meeting  at  one  point.  The  point  where  the  planes  and 
intersections  all  meet,  is  called  the  vertex  of  the  trie- 
dral. The  intersections  are  its  edges,  and  the  planes 
are  its  faces. 

The  corners  of  a  room,  or  of  a  chest,  are  illustrations 
of  triedrals  with  rectangular  faces.  The  point  of  a  tri- 
angular file,  or  of  a  small-sword,  has  the  form  of  a 
triedral  with  acute  faces. 


TRIEDRALS. 


197 


The  triedral  has  many  things  analogous  to  the  plane 
triangle.  It  has  been  called  a  solid  triangle;  and  more 
frequently,  but  with  less  propriety,  a  solid  angle.  The 
three  faces,  combined  two  and  two,  make  three  diedrals, 
and  the  three  intersections,  combined  two  and  two,  make 
three  plane  angles.  These  six  are  the  six  elements  or 
principal  parts  of  a  triedral. 

Each  face  is  the  plane  of  one  of  the  plane  angles,  and 
two  faces  are  said  to  be  equal  when  these  angles  are  equal. 

Two  triedrals  are  said  to  be  equal  when  their  several 
planes  may  coincide,  without  regard  to  the  extent  of  the 
planes.  Since  each  plane  is  determined  by  two  lines, 
it  is  evident  that  two  triedrals  are  equal  Avhen  their 
several  edges  respectively  coincide. 

583.  A  triedral  which  has  one  rectangular  diedral, 
that  is,  whose  measure  is  a  right  angle,  is  called  a  rect- 
angular triedral.  If  it  has  two,  it  is  hirectangular ;  if  it 
has  three,  it  is  trirecf angular. 

.A  triedral  which  has  two  of  its  faces  equal,  is  called 
isosceles;  if  all  three  are  equal,  it  is  equilateral. 

SYMMETRICAL    TRIEDRALS. 


584.  If  the  edges  of  a  triedral  be  produced  beyond 
the  vertex,  they  form  the  edges 
of  a  new  triedral.  The  faces  of 
these  two  triedrals  are  respect- 
ively equal,  for  the  angles  are 
vertical. 

Thus,  the  angles  ASC  and  ESD 
are  equal ;  also,  the  angles  BSC 
and  FSE  are  equal,  and  the  an- 
gles ASB  and  DSF. 

The  diedrals  whose  edgres   are  FS  and   BS   are  also 


198  ELEMENTS    OF    GEOMETRY. 

equal,  since,  being  formed  by  the  same  planes,  EFSBC 
and  DFSBA,  they  are  vertically  opposite  diedrals  (555). 
The  same  is  true  of  the  diedrals  whose  edges  are  DS  and 
SA,  and  of  the  diedrals  whose  edges  are  ES  and  SC. 

In  the  diagram,  suppose  ASB  to  be  the  plane  of  the 
paper,  C  being  above  and  E  below  that  plane. 

But  the  two  triedrals  are  not  equal,  for  they  can  not 
be  made  to  coincide,  although  composed  of  parts  which 
are  respectively  equal.  This  will  be  more  evident  if  the 
student  will  imagine  himself  within  the  first  triedral, 
his  head  toward  the  vertex,  and  his  back  to  the  plane 
ASB.  Then  the  plane  ASC  will  be  on  the  right  hand, 
and  BSC  on  the  left.  Then  let  him  imagine  himself  in 
the  other  triedral,  his  head  toward  the  vertex,  and  his 
back  to  the  plane  FSD,  which  is  equal  to  ASB.  Then 
the  plane  on  the  right  will  be  FSE,  which  is  equal  to 
BSC,  the  one  that  had  been  on  the  left;  and  the  plane 
now  on  the  left  will  be  DSE,  equal  to  the  one  that  had 
been  on  the  right. 

Now,  since  the  equal  parts  are  not  similarly  situated, 
the  two  figures  can  not  coincide. 

Then  the  difference  between  these  two  triedrals  con- 
sists in  the  opposite  order  in  which  the  parts  are  ar- 
ranged. ■  This  may  be  illustrated  by  two  gloves,  which 
we  may  suppose  to  be  composed  of  exactly  equal  parts. 
But  they  arc  arranged  in  reverse  order.  The  right 
hand  glove  will  not  fit  the  left  hand.  The  tAvo  hands 
themselves  are  examples  of  the  same  kind. 

5^5,  When  two  magnitudes  are  composed  of  parts 
respectively  equal,  but  arranged  in  reverse  order,  they 
are  said  to  be  symmetrical  magnitudes. 

The  word  symmetrical,  as  here  used,  has  essentially 
the  same  meaning  as  that  given  in  Plane  Geometry  (158). 
Two  symmetrical  plane  figures,  or  parts  of  a  figure,  are 


TRIEDRALS.  199 

divided  by  a   straight  line,  while   two   such   figures   in 
space  are  divided  by  a  plane. 

When  two  plane  figures  are  symmetrical,  they  are  also 
equal,  for  one  can  be  turned  over  to  coincide  with  the 
other,  as  with  the  figures  m  and  n  in  Article  282.  But 
this  is  not  possible,  as  just  shown,  with  figures  that  are 
not  in  one  plane. 

ANGLES    OF   A    TRIEDRAL. 

586.  Theorem. — Each  plane  angle  of  a  triedral  is  less 
than  the  sum  of  the  other  iivo. 

The  theorem  is  demonstrated,  when  it  is  shown  that 
the  greatest  angle  is  less  than  the  sum  of  the  oth^r  two. 

Let  ASB  be  the  largest  of  the  three  angles  of  the 
triedral  S.  Then,  from  the 
angle  ASB  take  the  part 
ASD,  equal  to  the  angle 
ASC.  Join  the  edges  SA 
and  SB  by  any  straight 
line  AB.  Take  SC  equal 
to  SD,  and  join  AC  and  BC. 

Since  the  triangles  ASD  and  ASC  are  equal  (284), 
AD  is  equal  to  AC.  But  AB  is  less  than  the  sum  of 
AC  and  BC,  anJ  from  these,  subtracting  the  equals  AD 
and  AC,  we  have  BD  less  than  BC.  Hence,  the  trian- 
gles BSD  and  BSC  have  two  sides  of  the  one  equal  to 
two  sides  of  the  other,  and  the  third  side  BD  less  than 
the  third  side  BC.  Therefore,  the  included  angle  BSD 
is  less  than  the  angle  BSC.  Adding  to  these  the  equal 
angles  ASD  and  ASC,  we  have  the  angle  ASB  less  than 
the  sum  of  the  angles  ASC  and  BSC. 

587.  Theorem. —  Tlie  sum  of  the  plane  angles  which 
form  a  triedral  is  always  less  than  four  right  angles. 


2D0  ELEMENTS    OF   GEOMETRY. 

Through  any  three  points,  one  in  each  edge  of  the 
triedral,  let  the  plane  ABC  pass,  making  the  intersec- 
tions AB,  BC,  and  AC,  with  the  faces. 

There  is  thus  formed  a  triedral  at  each  of  the  points 
A,  B,  and  C.     Then    the  angle   BAC   is   less   than   the 
sum  of  BAS  and  CAS  (586).     The  angle  ABC  is  less 
than  the  sum  of  ABS  and 
CBS.     The   angle  BCA  is 
less  than  the  sum  of  ACS         J^f^trTT^... 
and  BCS.    Adding  together  "~~~^- 

these   inequalities,  we   find 
that  the  sum  of  the  angles 

of  the  triangle  ABC,  which  is  two  right  angles,  is  less 
than  the  sum  of  the  six  angles  at  the  bases  of  the  tri- 
angles on  the  faces  of  the  triedral  S. 

The  sum  of  all  the  angles  of  these  three  triangles  is  six 
ricrht  anojles.  Therefore,  since  the  sum  of  those  at  the 
bases  is  more  than  two  right  angles,  the  sum  of  those 
at  the  vertex  S  must  be  less  than  four  right  angles. 

5SS.  To  assist  the  student  to  understand  this  theo- 
rem, let  him  take  any  three  points  on  the  paper  or 
blackboard  for  A,  B,  and  C.  Take  S  at  some  distance 
from  the  surface,  so  that  the  plane  angles  formed  at  S 
w^ill  be  quite  acute.  Then  let  S  approach  the  surface 
of  the  triangle  ABC.  Evidently  the  angles  at  S  be- 
come larger  and  larger,  until  the  point  S  touches  the 
surface  of  the  triangle,  when  the  sum  of  the  angles 
becomes  four  right  angles,  and,  at  the  same  time,  the 
triedral  becomes  one  plane. 

SUPPLEMENTARY    TRIEDRALS. 

589.  Theorem If,  from  a  point  ivithin  a  triedral, 

perpendicular  lines  fall  on   the  several  faces,  these  lines 


TRIEDRALS. 


201 


will  he  the  edges  of  a  second  triedral,  whose  faces  will  be 
supplements  respectively  of  the  diedrals  of  the  first;  and 
the  faces  of  the  first  will  be  respectively  supplements  of  tht 
diedrals  of  the  second  triedral. 

A  plane  angle  is  not  strictly  the  supplement  of  a  die- 
dral,  but  we  understand,  by  this  abridged  expression, 
that  the  plane  angle  is  the  supplement  of  that  which 
measures  the  diedral. 

If  from  the  point  E,  within  the  triedral  ABCD,  the 
perpendiculars  EF,  EG,  and  EH 
fall  on  the  several  faces,  then 
these  lines  form  a  second  trie- 
dral, whose  faces  are  FEH,  FEG, 
and  GEH. 

Then  the  angle  FEH  is  the 
supplement  of  the  diedral  whose 
edge  is  BA,  for  the  sides  of  the 
angle  are  perpendicular  to  the 
faces  of  the  diedral  (559).     For 

the  same  reason,  the  angle  FEG  is  the  supplement  rjf 
the  diedral  whose  edge  is  CA,  and  the  angle  GEH  is  the 
supplement  of  the  diedral  whose  edge  is  DA. 

But  it  may  be  shown  that  these  two  triedrals  have  a 
reciprocal  relation;  that  is,  that  the  property  just  proved 
of  the  second  toward  the  first,  is  also  true  of  the  first 
toward  the  second. 

Let  BF  and  BH  be  the  intersections  of  the  face  FEH 
with  the  faces  BAG  and  BAD ;  CF  and  CG  be  the  inter- 
sections of  the  face  FEG  with  the  faces  BAG  and  CAD ; 
and  DG  and  DH  be  the  intersections  of  the  face  GEH 
with  the  faces  GAD  and  BAD. 

Now,  since  the  plane  FBH  is  perpendicular  to  each 
of  the  planes  BAG  and  BAD  (556),  their  intersection 
AB  is  perpendicular  to  the  plane  FBH  (558).     For  a 


202  ELEMENTS  OF  GEOMETRY. 

like  reason,  AC  is  perpendicular  to  the  plane  FCG  and 
AD  is  perpendicular  to  the  plane  GDH.  Then,  reason- 
ing as  above,  we  prove  that  the  angle  BAG  is  the  sup- 
plement of  the  diedral  whose  edge  is  FE;  and  that  each 
of  the  other  faces  of  the  first  triedral  is  a  supplement 
of  a  diedral  of  the  second. 

590.  Two  triedrals,  in  which  the  faces  and  diedral 
angles  of  the  one  are  respectively  the  supplements  of 
the  diedral  angles  and  faces  of  the  other,  are  called 
supplementary  triedrals. 

Instead  of  placing  supplementary  triedrals  each  within 
the  other,  as  above,  they  may  be  supposed  to  have  their 
vertices  at  the  same  point.  Thus,  at  the  point  A,  erect 
a  perpendicular  to  each  of  the  three  faces  of  the  trie- 
dral ABCD,  and  on  the  side  of  the  face  toward  the 
triedral.  A  second  triedral  is  thus  formed,  which  is 
supplementary  to  the  triedral  ABCD,  and  is  symmet- 
rical to  the  one  formed  within. 


SUM    OF    THE    DIEDEALS, 

«>91.  Theorem. — In  every  triedral  the  sum  of  tJie  three 
diedral  angles  is  greater  than  two  right  angles^  and  less 
than  six. 

Consider  the  supplementary  triedral,  with  the  given 
one.  Now,  the  sum  of  the  three  diedrals  of  the  given 
triedral,  and  of  the  three  faces  of  its  supplementary  tri- 
edral, .  must  be  six  right  angles ;  for  the  sum  of  each 
pair  is  two  right  angles.  But  the  sum  of  the  faces  of 
the  supplementary  triedral  is  less  than  four  right  angles 
(587),  and  is  greater  than  zero.  Subtracting  this  sum 
from  the  former,  the  remainder,  being  the  sum  of  the 
three  diedrals  of  the  given  triedral,  is  greater  than  two 
and  less  than  six  right  angles. 


TRIEDRALS.  203 

EQUALITY    OF    TRIEDRALS. 

502.  Theorem. —  When  two  triedrals  have  two  faces^ 
and  the  included  diedral  of  the  one  respectively  equal  to 
the  corresponding  p>arfs  of  the  other,  then  the  remaining 
face  and  diedrals  of  the  first  are  respectively  equal  to  tk3 
correspondiyig  parts  of  the  other. 

There  are  tAvo  cases  to  be  considered. 

1st.  Suppose  the  angles  AEO  and  BCG  equal,  and 
the  angles  AEI 
and  BCD  equal, 
also  the  included 
diedrals  whose 
edges  are  AE  and 
BC.  Let  the  ar- 
rangement be  the 
same  in  both,  so 
that,    if     we     go 

around  one  triedral  in  the  order  0,  A,  I,  0,  and  around 
the  other  in  the  order  G,  B,  D,  G,  in  both  cases  the 
triedral  will  be  on  the  right.  Then  it  may  be  proved 
that  the  two  triedrals  are  equal. 

Place  the  angle  BCD  directly  upon  its  equal,  AEI. 
Since  the  diedrals  are  equal,  and  are  on  the  same  side 
of  the  plane  AEI,  the  planes  BCG  and  AEO  will  coin- 
cide. Since  the  angles  BCG  and  AEO  are  equal,  the 
lines  CG  and  EO  will  coincide.  Thus,  the  angles 
DCG  and  lEO  coincide,  and  the  two  triedrals  coincide 
throughout. 

2J.  Let  the  angles  AEO  and  DCG  be  equal,  and  the 
angles  AEI  and  BCD,  also  the  included  diedrals,  whose 
edges  are  AE  and  DC.  But  let  the  arrangement  be  re- 
verse ;  that  is,  if  we  go  around  one  triedral  in  the  order 
0,  A,  I,  0,  and  around  the  other  in  the  order  G,  D,  B,  G, 


204 


ELEMENTS    OF    GEOMETRY. 


h:^:'-' 


,j^._ 


7F 


/ 


in  one  case  the  triedral  will  be  to  the  right,  and  in  the 
other  it  will  be  to  the  left  of  us.  Then  it  may  be 
proved  that  the  two  triedrals  are  symmetrical. 

One  of  the  triedrals  can  be  made  to  coincide  with  the 
symmetrical  of  the  other ;  for  if  the  edges  BC,  GO,  and 
DC  be  produced  beyond  C,  the  triedral  CFHK  will 
have  two  faces 
and  the  included 
diedral  respect- 
ively equal  to 
those  parts  of  the 
triedral  EAOI, 
and  arranged  in 
the  same  order ; 
that  is,  the  re- 
verse of  the  tri- 
edral CDGB. 
Hence,  as  just 
shown,  the  trie- 
drals CFHK  and 
EAOI  are  equal. 
Therefore,  EAOI  and  CDGB  are  symmetrical  triedrals. 

In  both  cases,  all  the  parts  of  each  triedral  are  re- 
spectively equal  to  those  of  the  other. 

59.^.  Theorem. —  When  hvo  triedrals  have  one  face  and 
(he  two  adjacent  diedrals  of  the  one  respectively  equal  to 
the  corresponding  parts  of  the  other,  then  the  remaining 
faces  and  diedral  of  the  first  are  respectively  equal  to 
the  corresponding  parts  of  the  other. 

Suppose  that  the  faces  AEI  and  BCD  are  equal,  that 
the  diedrals  whose  edges  are  AE  and  BC  are  equal,  that 
the  diedrals  whose  edges  are  IE  and  DC  are  equal,  and 
that  these  parts  are  similarly  arranged  in  the  two  trie- 
drals.    Then  the  one  may  coincide  with  the  other. 


TRIEDRALS. 


205 


Therefore,  it 


For  BCD  may  coincide  with  its  equal  AEI,  BC  fall- 
ing on  AE.  Then  the  plane  of  BCG  must  coincide 
with  that  of  AEO,  since  the  diedrals  are  equal ;  and  the 
line  CG  will  fall  in  the  plane  of  AEO.  For  a  similar 
reason  CG  will  fall  on  the  plane  of  lEO. 
must  coincide  with 
their  intersection 
EO,  and  the  two 
triedrals  coincide 
throughout. 

When  the  equal 
parts  are  in  re- 
verse order  in  the 
two  triedrals,  the 

arrangement  in  one  must  be  the  same  as  in  the  sym- 
metrical of  the  other.  Therefore,  in  that  case,  the  two 
triedrals  would  be  symmetrical. 

In  both  cases,  all  the  parts   of  each  triedral  are  re- 
spectively equal  to  those  of  the  other. 

594.  Theorem. — A71  isosceles  triedral  and  its  symmet- 
rical are  equal. 

Let  ABCD  be  an  isosceles  triedral,  having  the  faces 
BAC  and  DAC  equal,  and  let  AEFG 
be  its  symmetrical  triedral. 

Now,  the  faces  BAC,  DAC,  FAG, 
and  FAE,  are  all  equal  to  each  other. 
The  diedrals  whose  edges  are  AC  and 
AF  being  vertical,  are  also  equal. 
Hence,  the  faces  mentioned  being  all 
equal,  corresponding  equal  parts  may 
be  taken  in  the  same  order  in  both 
triedrals ;  that  is,  the  face  EAF  equal 
to  the  face  BAC,  and  the  face  FAG 
equal  to  CAD.     Therefore,  the  two  triedrals  are  equal. 


206 


ELEMENTS  OF  GEOMETRY. 


395«  Corollary — In  an  isosceles  triedral,  the  diedrals 
opposite  the  equal  faces  are  equal.  For  the  diedrals 
whose  edges  are  AB  and  AD,  are  eacl:  equal  to  the 
diedral  whose  edge  is  AE. 

50l>.  Corollary Conversely,  if  in  any  triedral  two 

of  the  diedral  angles  are  equal,  then  the  faces  opposite 
tiiese  diedrals  are  equal,  and  the  triedral  is  isosceles. 
For,  as  in  the  above  theorem,  the  given  triedral  can  be 
shown  to  be  equal  to  its  symmetr'iCal. 

507.  Theorem. —  When  i.wo  triedrals  have  two  faces  of 
the  one  respectively/  equal  to  two  faces  of  the  other,  and 
the  included  diedrals  unequal,  then  the  third  faces  are 
unequal,  and  that  face  is  greater  which  is  ojyposite  the 
greater  diedral. 

Suppose  that  the  faces  CBD  and  EAi  are  equal,  and 


that  the  faces  CBF  and  EAO  are  also  equal,  but  that 
the  diedral  whose  edge  is  CB  is  greater  than  the  die- 
dral whose  edge  is  EA.  Then  the  face  FBD  will  be 
greater  than  the  face  OAI. 

Through  the  line  BC,  let  a  plane  GBC  pass,  making 
with  the  plane  DBC  a  diedral  equal  to  that  whose  edge 
is  AE.  In  this  plane,  make  the  angle  CBG  equal  to 
EAO.     Let  the  diedral  FBCG  be  bisected  by  the  plane 


TRIEDRALS.  207 

IIBC.  BH  being  the  intersection  of  this  plane  with  the 
pbine  FBD. 

Then  the  two  triedrals  BCDG  and  AEIO,  having  two 
faces  and  the  included  diedral  in  the  one  equal  to  the 
corresponding  parts  in  the  other,  will  have  the  remain- 
ing parts  equal.  Hence,  the» faces  DBG  and  lAO  aie 
equal. 

Again,  the  two  triedrals  BCFH  and  BCGH  have  the 
faces  CBF  and  CBG  equal,  by  construction,  the  face 
CBH  common,  and  the  included  diedrals  equal,  by  con- 
struction. Therefore,  the  third  faces  FBH  and  GEH 
are  equal. 

To  each  of  these  equals  add  the  face  HBD,  and  we 
have  the  face  FBD  equal  to  the  sum  of  GBH  and  HBD. 
But  in  the  triedral  BDGH,  the  face  DBG  is  less  thai: 
the  sum  of  the  other  two  faces,  GBH  and  HBD  (586) 
Hence,  the  face  DBG  is  less  than  FBD.  Therefore,  the 
face  OAI,  equal  to  DBG,  is  less  than  FBD. 

598.  Corollary — Conversely,  when  two  triedrals  have 
two  faces  of  the  one  respectively  equal  to  two  faces  of 
the  other,  and  the  third  faces  are  unequal,  then  the  die- 
dral opposite  the  greater  face  is  greater  than  the  diedral 
opposite  the  less. 

599.  Theorem. —  When  two  triedrals  have  their  three 
faces  resj)ectively  equal,  their  diedrals  will  he  respectively 
equal;  and  the  two  triedrals  are  either  equal,  or  they  are 
symmetrical. 

When  two  faces  of  one  triedral  are  respectively  equal 
to  those  of  another,  if  the  included  diedrals  are  une- 
qual, then  the  opposite  faces  are  unequal  (597).  But, 
by  the  hypothesis  of  this  theorem,  the  third  faces  are 
equal.  Therefore,  the  diedrals  opposite  to  those  faces 
must  be  equal. 

In  the  same  manner,  it  may  be  shown  that  the  other 


203  ELEMENTS   OF   GEOMETRY. 

diedral  angles  of  the  one,  are  equal  to  the  corresponding 
diedral  angles  of  the  other  triedral.  Therefore,  the  trie- 
drals  are  either  equal  or  symmetrical,  according  to  the 
arrangement  of  their  parts. 

600.  Theorem Tivo  triedrah  which  have  their  die- 

drals  respectively  equal,  hmoe  also  their  faces  resjyectively 
equal ;  and  the  two  triedrals  are  either  equal,  or  they  are 
symmetrical. 

Consider  the  supplementary  triedrals  of  the  two  given 
triedrals.  These  will  have  their  faces  respectively  equal, 
because  they  are  the  supplements  of  equal  diedral  an- 
gles (589).  Since  their  faces  are  equal,  their  diedrals 
are  equal  (599).  Then  the  two  given  triedrals,  having 
^heir  faces  the  supplements  of  these  equal  diedrals,  will 
nave  those  faces  equal ;  and  the  triedrals  are  either 
equal  or  symmetrical,  according  to  the  arrangement  of 
their  parts. 

601.  The  student  may  notice,  in  every  other  case  of 
equal  triedrals,  the  analogy  to  a  case  of  equality  of  tri- 
angles; but  the  theorem  just  demonstrated  has  nothing 
analogous  in  plane  geometry. 

60!3.  Corollary.  —  All  trirectangular  triedrals  are 
equal. 

603.  Corollary. — In  all  cases  where  two  triedrals  are 
either  equal  or  supplementary,  equal  faces  are  opposite 
equal  diedral  angles 


EXERCISES. 

694. — 1.   In  any  triedral,  the  greater  of  two  faces  is  opposite 
to  the  greater  diedral  angle;  and  conversely. 

2.  Demonstrate   the    principles  stated    in    the  last  sentence  of 
Article  590. 

3.  If  a  triedral  have  one  riglit  diedral  angle,  then  an  adjacent 


POLYEDRALS.  209 

fac3  and  its  opposite  diedral  are  either  both  acute,  both  right,  or 
Vioth  obtuse. 


POLYEDRALS. 

605.  A  PoLYEDRAL  is  the  figure  formed  by  several 
planes  which  meet  at  one  point.  Thus,  a  polyedral  is 
composed  of  several  angles  having  their  vertices  at  a 
common  point,  every  edge  being  a  side  of  two  of  the 
angular  faces.  The  triedral  is  a  polyedral  of  three 
faces. 

600.  Problem Any  polyedral   of  more   than   three 

faces  may  he  divided  into  triedrah 

For  a  plane  may  pass  through  any  two  edges  which 
are  not  adjacent.  Thus,  a  polyedral  of  four  faces  may 
be  divided  into  two  triedrals ;  one  of  five  faces,  into 
three ;  and  so  on. 

GOT.  This  is  like  the  division  of  a  polygon  into  tri- 
angles.    The  plane  passing  through  two  edges  not  adja- 
cent is  called  a  diag- 
07ial  plane.  , 

A    polyedral     is  //i\ 

called   convex,  when  //  i    y^  / 

every  possible  diag-  /kC '      \\         A''' 

onal  plane  lies  within       /     /  ^~^\\A     /  ^ 
the  figure;  otherwise  '  I  ^  \ 

it  is  called  concave. 

60S.  Corollary If  the  plane  of  one  face  of  a  con- 
vex polyedral  be  produced,  it  can  not  cut  the  polyedral. 

609.  Corollary. — A  plane  may  pass  through  the  ver- 
tex of  a  convex  polyedral,  without  cutting  any  face  of 
the  polyedral. 

610.  Corollary — A  plane  may  cut  all  the  edges  of  a 
convex  polyedral.     The  section  is  a  convex  polygon. 

Geom.— 18 


210  ELEMENTS  OF   GEOMETRY. 

Gil.  When  any  figure  is  cut  by  a  plane,  the  figure 
that  is  defined  on  the  plane  by  the  limits  of  the  figure 
so  cut,  is  called  a  plane  section. 

Several  properties  of  triedrals  are  common  to  other 
polyedrals. 

612.  Theorem — The  sum  of  all  the  angles  of  a  convex 
polyedral  is  less  than  four  right  angles. 

For,  suppose  the  polyedral  to  be  cut  by  a  plane,  then 
the  section  is  a  polygon  of  as  many  sides  as  the  polye- 
dral has  faces.  Let  n  represent  the  number  of  sides  of 
the  polygon.  The  plane  cuts  off  a  triangle  on  each  face 
of  the  polyedral,  making  n  triangles.  Now,  the  sum  of 
the  angles  of  this  polygon  is  2n — 4  right  angles  (424), 
and  the  sum  of  the  angles  of  all  these  triangles  is  2n 
right  angles.  Let  v  right  angles  represent  the  sum  of 
the  angles  at  the  vertex  of  the  polyedral ;  then,  2n  right 
angles  being  the  sum  of  all  the  angles  of  the  triangles, 
2n  —  V  is  the  sum  of  the  angles  at  their  bases. 

Now,  at  each  vertex  of  the  polygon  is  a  triedral  hav- 
ing an  angle  of  the  polygon  for  one  fjice,  and  angles  at 
the  bases  of  the  triangles  for  the  other  two  faces. 
Then,  since  two  faces  of  a  triedral  are  greater  than  the 
third,  the  sum  of  all  the  angles  at  the  bases  of  the  tri- 
angles is  greater  than  the  sum  of  the  angles  of  the 
polygon.     That  is, 

2n  —  -y  >  2n  —  4. 

Adding  to  both  members  of  this  inequality,  v  +  4,  and 
s  ibtracting  2n,  we  have  4  ^  v.  That  is,  the  sum  of  the 
angles  at  the  vertex  is  less  than  four  right  angles. 

This  demonstration  is  a  generalization  of  that  of 
Article  587.  The  student  should  make  a  diagram  and 
special  demonstration  for  a  polyedral  of  five  or  six 
faces. 


DESCRIPTIVE    GEOMETRY.  211 

013.  Theorem. — In  any  convex  poJi/edral,  the  sum  of 
the  diedrals  is  greater  than  the  sum  of  the  angles  of  a 
polygon  having  the  same  number  of  sides  that  the  poly- 
edral  has  faces. 

Let  the  given  polyedral  be  divided  by  diagonal  planes 
into  triedrals.  Then  this  theorem  may  be  demonstrated 
like  the  analogous  proposition  on  polygons  (423).  The 
remark  made  in  Article  346  is  also  applicable  here. 

DESCRIPTIVE    GEOMETRY. 

614.  In  the  former  part  of  this  work,  we  have  found 
problems  in  drawing  to  be  the  best  exercises  on  the 
principles  of  Plane  Geometry.  At  first  it  appears  im- 
possible to  adapt  such  problems  to  the  Geometry  of 
Space ;  for  a  drawing  is  made  on  a  plane  surface,  while 
the  figures  here  investigated  are  not  plane  figures. 

This  object,  however,  has  been  accomplished  by  the 
most  ingenious  methods,  invented,  in  great  part,  by 
Monge,  one  of  the  founders  of  the  Polytechnic  School 
at  Paris,  the  first  who  reduced  to  a  system  the  elements 
of  this  science,  called  Descriptive  Geometry. 

Descriptive  Geometry  is  that  branch  of  mathemat- 
ics which  teaches  how  to  represent  and  determine,  by 
means  of  drawings  on  a  plane  surface,  the  absolute  or 
relative  position  of  points  or  magnitudes  in  space.  It 
is  beyond  the  design  of  the  present  work  to  do  more 
than  allude  to  this  interesting  and  very  useful  science. 

EXERCISES. 

615. — 1.  What  is  the  locus  of  those  points  in  space,  each  of 
which  is  equally  distant  from  three  given  points? 

2.  AVhat  is  the  locns  of  those  points  in  space,  each  of  which  is 
equally  distant  from  two  given  planes? 


212  ELEMENTS    OF    GEOMETRY. 

3.  What  is  the  locus  of  those  points  in  space,  eacli  of  which 
is  equally  distant  from  three  given  planes? 

4.  What  is  the  locus  of  those  poin*  in  space,  each  of  which 
is  equally  distant  from  two  given  straight  lines  which  lie  in  the 
same  plane  ? 

5.  What  is  the  locus  of  those  points  in  space,  each  of  which 
is  equally  distant  from  three  given  straight  lines  which  lie  in  the 
same  plane? 

6.  What  is  the  locus  of  those  points  in  space,  such  that  the 
sum  of  the  distances  of  each  from  two  given  planes  is  equal  to  a 
given  straight  line  ? 

7.  If  each  diedral  of  a  triedral  be  bisected,  the  three  planes 
have  one  common  intersection. 

8.  If  a  straight  line  is  perpendicular  to  a  plane,  every  plane 
parallel  to  the  given  line  is  perpendicular  to  the  given  plane. 

9.  Given  any  two  straight  lines  in  space;  either  one  plane  may 
pass  through  both,  or  two  parallel  planes  may  pass  through  them 
respectively. 

10.  In  the  second  case  of  the  preceding  exercise,  a  line  which 
is  perpendicular  to  both  the  given  lines  is  also  perpendicular  to 
the  two  planes. 

11.  If  one  face  of  a  triedral  is  rectangular,  then  an  adjacent 
diedral  angle  and  its  opposite  face  are  either  both  acute,  both 
right,  or  both  obtuse. 

12.  Apply  to  planes,  diedrals,  and  triedrals,  respectively,  such 
properties  of  straight  lines,  angles,  and  triangles,  as  have  not 
already  been  stated  in  this  chapter,  determining,  in  each  case, 
whether  the  principle  is  true  when  so  applied. 


TETKAEDRONS.  213 


CHAPTER    X. 

POLYEDRONS. 

OlO.  A  PoLYEDRON  is  a  solid,  or  portion  of  space, 
bounded  by  plane  surfaces.  Each  of  these  surfaces  is 
a  face^  their  several  intersections  are  edge^^  and  the 
points  of  meeting  of  the  edges  are  vertices  of  the  poly- 
edron. 

617.  Corollary — The  edges  being  intersections  of 
planes,  must  be  straight  lines.  It  follows  that  the 
faces  of  a  polyedron  are  polygons. 

618.  A  Diagonal  of  a  polyedron  is  a  straight  line 
joining  two  vertices  which  are  not  in  the  same  face. 

A  Diagonal  Plane  is  a  plane  passing  through  three 
vertices  which  are  not  in  the  same  face. 

TETKAEDRONS. 

619.  We  have  seen  that  three  planes  can  not  inclose 
a  space    (581).       But    if   any 

point  be  taken  on  each  edge 
of  a  tricdral,  a  plane  passing 
through    these    three     points 

would,  with  the  three  faces  of  /  \^4   \ 

the  triedral,  cut  off  a  portion        ^m^^  '^     \ 

of  space,  which  would  be  in- 
closed by  four  triangular  faces. 

A  Tetraedron  is  a  polyedron  having  four  faces. 


214  ELEMENTS  OF  GEOMETRY. 

0!S0.  Problem. — Any  four  points  whatever ^  which  do 
not  all  lie  in  one  plane,  may  he  taken  as  the  four  vertices 
of  a  tetraedron. 

For  they  may  be  joined  two  and  two,  by  straight 
lines,  thus  forming  the  six  edges  ;  and  these  bound  the 
four  triangular  faces  of  the  figure. 

031.  Either  face  of  the  tetraedron  may  be  taken  as 
the  base.  Then  tlie  other  faces  are  called  the  sides,  the 
vertex  opposite  the  base  is  called  the  vertex  of  the 
tetraedron,  and  the  altitude  is  the  perpendicular  distance 
from  the  vertex  to  the  plane  of  the  base.  In  some 
cases,  the  perpendicular  falls  on  the  plane  of  the  base 
produced,  as  in  triangles. 

622.  Corollary — If  a  plane  parallel  to  the  base  of  a 
tetraedron  pass  through  the  vertex,  the  distance  between 
this  plane  and  the  base  is  the  altitude  of  the  tetrae- 
dron (574). 

62-^.  Theorem. —  There  is  a  point  equally  distant  from 
the  four  vertices  of  any  tetraedron. 

In  the  plane  of  the  face  BCF,  suppose  a  circle  w^hose 
circumference  passes  through 
the  three  points  B,  C,  and  F. 
At  the  center  of  this  circle, 
erect  a  line  perpendicular  to 
the  plane  of  BCF. 

Every  .  point  of  this  per- 
pendicular is  equally  distant 
from  the  three  points  B,  C, 
and  F  (531). 

In  the  same  manner,  let  a  line  perpendicular  to  the 
plane  of  BDF  be  erected,  so  that  every  point  shall  be 
equally  distant  from  the  points  B,  D,  and  F. 

These  two  perpendiculars  both  lie  in  one  plane,  the 
plane  which  bisects  the  edge  BF  perpendicularly  at  its 


TETRAEDRONS.  215 

center  (520).  These  two  perpendiculars  to  two  oblique 
planes,  being  therefore  oblique  to  each  other,  will  meet 
at  some  point.  This  point  is  equally  distant  from  the 
four  vertices  B,  C,  D,  and  F. 

634.  Corollary. — The  six  planes  which  bisect  perpen- 
dicularly the  several  edges  of  a  tetraedron  all  meet  in 
one  point.  But  this  point  is  not  necessarily  within  the 
tetraedron. 

63«5.  Theorem. —  There  is  a  point  within  every  tetrae- 
dron which  is  equally  distant  from  the  several  faces. 

Let  AEIO   be   any   tetraedron,   and  let   OB   be    the 
straight  line  formed  by  the 
intersection    of  two  planes,  A 

one    of    which    bisects    the  //  \ 

diedral  angle  whose  edge  is  /^    /       \ 

AO,  and  the  other  the  die-        ^y^  -— /— -^ .\^^ 

dral  whose  edge  is  EG.  \^   / ^^^.^^^''^ 

Now,  every  point  of  the  I 

first  bisecting  plane  is  equally 

distant  from  the  faces  lAO  and  EAO  (560) ;  and  every 
point  of  the  second  bisecting  plane  is  equally  distant 
from  the  faces  EAO  and  EIO.  Therefore,  every  point 
of  the  lino  BO,  which  is  the  intersection  of  those  bisect- 
ing planes,  is  equally  distant  from  those  three  faces. 

Then  let  a  plane  bisect  the  diedral  whose  edge  is  EI, 
and  let  C  be  the  point  where  this  plane  cuts  the  line  BO. 

Since  every  point  of  this  last  bisecting  plane  is  equally 
distant  from  the  faces  EAI  and  EOI,  it  follows  that  the 
point  C  is  equally  distant  from  the  four  faces  of  the  tet- 
raedron. Since  all  the  bisecting  planes  are  interior, 
the  point  found  is  within  the  tetraedron. 

626.  Corollary. — The  six  planes  which  bisect  the 
several  diedral  angles  of  a  tetraedron  all  meet  at  one 
point. 


216  ELEMENTS    OF   GEOMETRY. 

EQUALITY    OF    TETRAEDRONS. 

G27.  Theorem. — Two  tetraedrons  are  equal  where  three 
faces  of  the  one  art,  respectively  equal  to  three  faces  of  the 
other,  and  they  are  similarly  arranged. 

For  the  three  sides  of  the  fourth  face,  in  one,  must 
be  equal  to  the  same  lines  in  the  other.  Hence,  the 
fourth  faces  are  equal.  Then  each  diedral  angle  in  the 
one  is  equal  to  its  corresponding  diedral  angle  in  the 
other  (599).  In  a  word,  every  part  of  the  one  figure  is 
equal  to  the  corresponding  part  of  the  other,  and  the 
equal  parts  are  similarly  arranged.  Therefore,  the  two 
tetraedrons  are  equal. 

028.  Corollary. — Two  tetraedrons  are  equal  when  the 
six  edges  of  the  one  are  respectively  equal  to  those  of 
the  other,  and  they  are  similarly  arranged. 

629.  Corollary. — Tavo  tetraedrons  are  equal  when  two 
faces  and  the  included  diedral  of  the  one  are  respect- 
ively equal  to  those  parts  of  the  other,  and  they  are 
similarly  arranged. 

630.  Corollary — Two  tetraedrons  are  equal  when  one 
face  and  the  adjacent  diedrals  of  the  one  are  respect- 
ively equal  to  those  parts  of  the  other,  and  they  are 
similarly  arranged. 

6S1.  When  tetraedrons  are  composed  of  equal  parts 
in  reverse  order,  they  are  symmetrical. 

MODEL    TETRAEDRON. 

632.  The  student  may  easily  construct  a  model  of  a  tetrae- 
dron  when  the  six  edges  are  given.  First,  with  three  of  the  edges 
which  are  sides  of  one  face,  draw  the  triangle,  as  ABC.  Then, 
on  each  side  of  this  first  triangle,  as  a  base,  draw  a  triangle  equal 
to  the   corresponding   face;    all   of  which  can  be  done,  for  the 


TETRAEDRONS.  217 

edges,  that  is,  the  sides  of  these  triangles,  are  given.  Theiij  cut 
out  the  whole  figure  from  the  pa- 
per and  carefully  fold  it  at  the 
lines  AB,  BC,  and  CA.  Since 
BF  is  equal  to  BD,  CF  to  CE, 
and  AD  to  AE,  the  points  F,  D, 
and  E  may  be  united  to  form  a 
vertex. 

In  this  way  models  of  various  forms  may  be  made  with  mor( 
accuracy  than  in  wood,  and  the  student  may  derive  much  hel] 
from  the  work. 

But  he  must  never  forget  that  the  geometrical  figure  exists 
only  as  an  intellectual  conception.  To  assist  him  in  this,  he 
should  strive  to  generalize  every  demonstration,  stating  the  argu- 
ment without  either  model  or  diagram,  as  in  the  demonstration 
last  given. 

To  construct  models  of  symmetrical  tetraedrons,  the  drawings 
may  be  equal,  but  the  folding  should,  in  the  one  case,  be  up,  and 
in  the  other,  down. 


SIMILAR    TETRAEDRONS. 

633.  Since  similarity  consists  in  having  the  same 
form,  so  that  every  difference  of  direction  in  one  of 
two  similar  figures  has  its  corresponding  equal  differ- 
ence of  direction  in  the  other,  it  follows  that  when  two 
polyedrons  are  similar,  their  homologous  faces  are  simi- 
lar polygons,  their  homologous  edges  are  of  equal  die- 
dral  angles,  and  their  homologous  vertices  are  of  equal 
polyedrals. 

634.  Theorem —  Whe7i  two  tetraedrons  are  similar,  any 
edge  or  other  line  in  the  one  is  to  the  homologous  line  in 
the  second,  as  any  other  line  in  the  first  is  to  its  homolo- 
gous line  in  the  second. 

If  the   proportion  to  be   proved  is  between  sides  of 
homologous  triangles,  it  follows  at  once  from  the  simi- 
larity of  the  triangles, 
(leom. — 19 


218 


ELEMENTS  OF  GEOMETRY. 


When  the  edges  taken  in  one  of  the  tetraedrons  are 
not  fi  'des  of  one  face ;  as, 

AE  :  BC  : :  10  :  DF, 

A 


then,  AE  :  BC  :  :  IE  :  CD,  as  just  proved, 

and  1 0  :  DF  :  :  IE  :  CD. 

Therefore,  AE  :  BC  : :  10  :  DF. 

Again,  suppose  it  is  to  be  proved  that  the  altitudes 
AK  and  BH  have  the  same  ratios  as  two  homologous 
edges.  AK  and  BH  are  perpendicular  lines  let  fall  from 
the  homologous  points  A  and  B  on  the  opposite  faces. 
From  K  let  the  perpendicular  KN  fall  upon  the  edge 
10.  Join  AN,  and  from  H  let  the  perpendicular  HG 
fall  upon  DF,  which  is  homologous  to  10.     Join  BG. 

Now,  the  planes  AKN  and  EIO  are  perpendicular  to 
each  other  (556),  and  the  line  IN  m  one  of  them  is, 
by  construction,  perpendicular  to  their  intersection  KN. 
Hence,  IN  is  perpendicular  to  the  plane  AKN  (557). 
Therefore,  the  line  AN  is  perpendicular  to  IN,  and  the 
diedral  whose  edge  is  10  is  measured  by  the  angle 
ANK.  In  the  same  way,  it  is  proved  that  the  diedral 
whose  edge  is  DF,  is  measured  by  the  angle  BGII. 
But  these  two  diedrals,  being  homologous,  are  equal, 
the  angles  ANK  and  BGH  are  equal,  and  the  right  an- 
gled triangles  AKN  and  BHG  are  similar.  Therefore, 
AK  :  BH  :  :  AN  :  BG. 


TETRAEDRONS.  21U 

Also,  the  right  angled  triangles  ANI  and  BGD  are 
similar,  since,  by  hypothesis,  the  ap'^les  AIN  and  BDG 
are  equal.     Hence, 

AI  :  BD  :  :  AN  :  BG. 

Therefore,       AK  :  BH  :  :   AI  :  BD. 

Thus,  by  the  aid  of  similar  triangles,  it  may  be  proved 
that  any  two  liomologous  lines,  in  two  similar  tetrae- 
drons,  have  the  same  ratio  as  two  homologous  edges. 

635.  Tliccrem Two    tetraedrons    are    similar   ivhen 

their  faces  are  respectively  similar  triangles,  and  are  simi- 
larly arranged. 

For  we  know,  from  the  similarity  of  the  triangles, 
that  every  line  made  on  the  surface  of  one  may  have 
its  homologous  line  in  the  second,  making  angles  equal 
to  those  made  by  the  first  line. 

If  lines  be  made  through  the  figure,  it  may  be  shown, 
by  the  aid  of  auxiliary  lines,  as  in  the  corresponding 
proposition  of  similar  triangles,  that  every  possible  an- 
gle in  the  one  figure  has  its  homologous  equal  angle  in 
the  other. 

The  student  may  draw  the  diagrams,  and  go  through 
the  details  of  the  demonstration. 

636.  If  the  similar  faces  were  not  arranged  similarly, 
but  in  reverse  order,  the  tetraedrons  would  be  symmet- 
rically similar. 

637.  Corollary — Two   tetraedrons   are   similar  when 
three  ftices  of  the  one  are  respectively  similar  to  thos 
of  the  other,  and  they  are  similarly  arranged.     For  th 
fourth  faces,  having  their  sides  proportional,  are  simi- 
lar also. 

638.  Corollary — Two  tetraedrons  are  similar  when 
two  triedral  vertices  of  the  one  are  respectively  equal 
to  two  of  the  other,  and  they  are  similarly  arranged* 


320  ELEMENTS   OF   GEOMETRY. 

630.  Corollary. — Two  tetraedrons  are  similar  when 
the  edges  of  one  are  respectively  proportional  to  those 
of  the  other,  and  they  are  similarly  arranged. 

G40.  Theorem The   areas  of  homologous  faces   of 

similar  tetraedrons  are  to  each  other  as  the  squares  of 
their  edges. 

This  is  only  a  corollary  of  the  theorem  that  the  areas 
of  similar  triangles  are  to  each  other  as  the  squares  of 
their  sides. 

641.  Corollary. — The  areas  of  homologous  faces  of 
similar  tetraedrons  are  to  each  other  as  the  squares  of 
any  homologous  lines. 

642.  Corollary — The  area  of  any  face  of  one  tetrae- 
dron  is  to  the  area  of  a  homologous  face  of  a  similar 
tetraedron,  as  the  area  of  any  other  face  of  the  first  is 
to  the  area  of  the  homologous  face  of  the  second. 

643.  Corollary.  —  The  area  of  the  entire  surface  of 
one  tetraedron  is  to  that  of  a  similar  tetraedron  as  the 
squares  of  homologous  lines. 

TETRAEDEONS    CUT    BY   A    PLANE. 

044.  Theorem. — If  a  plane  cut  a  tetraedron  parallel 
to  the  base,  the  tetraedron  cut  off  is  similar  to  the  whole. 

For  each  triangular  side  is  cut  by  a  line  parallel  to 
its  base  (572),  thus  making  all  the  edges  of  the  two 
tetraedrons  respectively  proportional. 

645.  Theorem If  two  tetraedrons,  having  the   same 

altitude  and  their  bases  on  the  same  plane,  are  cut  by  a 
plane  parallel  to  their  bases,  the  areas  of  the  sections  will 
have  the  same  ratio  as  the  areas  of  the  bases. 

If  a  plane  parallel  to  the  bases  pass  through  the  ver- 
tex A,  it  will  also  pass  through  the  vertex  B  (022).     But 


TEirvAEDRONS.  221 

such  a  plane  is  parallel  to  the  cutting  plane  GHP  (566). 


Therefore,   the   tetraedrons  AGHK    and    BLNP    have 
equal  altitudes. 

The  tetraedrons  AEIO  and  AGHK  are  similar  (644). 
Therefore,  EIO,  the  base  of  the  first,  is  to  GHK,  the 
base  of  the  second,  as  the  square  of  the  altitude  of  the 
first  is  to  the  square  of  the  altitude  of  the  second  (641). 
For  a  like  reason,  the  base  CDF  is  to  the  base  LNP  as 
the  square  of  the  greater  altitude  is  to  the  square  of 
the  less. 

Therefore,  EIO  :  GHK  : :  CDF  :  LNP. 

By  alt3rnation, 

EIO  :    CDF   :  :  GHK  :  LNP. 

046.  Corollary — When  the  bases  are  equivalent  the 
sections  are  equivalent. 

647.  Corollary — When  the  bases  are  equal  the  sec- 
tions are  equal.     For  they  are  similar  and  equivalent. 

REGULAR    TETRAEDRON. 

648.  There  is  one  form  of  the  tetraedron  which  de- 
serves particular  notice.  It  has  all  its  faces  equilateral. 
This  is  called  a  regular  tetraedron. 

649.  Corollary — It  follows,  from  the  definition,  that 


222  ELEMENTS    OF    GEOMETRY. 

the  faces,  are  equal  triangles,  the  vertices  are  of  equal 
triedrals,  and  the  edges  are  of  equal  diedral  angles. 

050.  The  area  of  the  surface  of  a  tetraedron  is  found 
by  taking  the  sum  of  the  areas  of  the  four  faces.  When 
two  or  more  of  them  are  equal,  the  process  is  shortened 
by  multiplication.  But  the  discussion  of  this  matter 
will  be  included  in  the  subject  of  the  areas  of  pyra- 
mids. 

The  investigation  of  the  measures  of  volumes  will  be 
given  in  another  connection. 

EXERCISES. 

651. — 1.  State  other  cases,  when  two  tetraedrons  are  similar, 
in  addition  to  those  given,  Articles  G35  to  639. 

2.  In  any  tetraedron,  the  lines  which  join  the  centers  of  the 
opposite  edges  bisect  each  other. 

3.  If  one  of  the  vertices  of  a  tetraedron  is  a  trirectangular  tri- 
edral,  the  square  of  the  area  of  the  opposite  face  is  equal  to  the 
sum  of  the  squares  of  the  areas  of  the  other  three  faces. 

PYRAMIDS. 

6,^2.  If  a  polyedral  is  cut  by  a  plane  which  cuts  its 
several  edges,  the  section  is  a  pol}/gon,  and  a  portion  of 
space  is  cut  oif,  which  is  called  a  pyramid. 

A  Pyramid  is  a  polyedron  having  for  one  face  any 


polygon,  and  for  its  other  faces,  triangles  whose  vertices 
meet  at  one  point. 


PYRAMIDS.  223 

The  polygon  is  the  base  of  the  pyramid,  the  triangles 
are  its  sides,  and  their  intersections  are  the  lateral  edges 
of  the  pyramid.  The  vertex  of  the  polyedral  is  the 
vertex  of  the  pyramid,  and  the  perpendicular  distance 
from  that  point  to  the  plane  of  the  base  is  its  altitude. 

Pyramids  are  called  triangular,  quadrangular,  pentn; - 
onal,  etc.,  according  to  the  polygon  which  forms  tic 
base.     The  tetraedron  is  a  triangular  pyramid. 

053.  Problem. — Every  pyramid  can  he  divided  into  the 
same  number  of  tetraedrons  as  its  base  can  be  into  triangles. 

Let  a  diagonal  plane  pass  through  the  vertex  of  the 
pyramid  and  each  diagonal  of  the  base,  and  the  solu- 
tion is  evident. 


EQUAL    PYRAMIDS. 

654.  Theorem — Two  pyramids  are  equal  tvJien  the  base 
and  two  adjacent  sides  of  the  07ie  are  respectively  equal  to 
the  corresponding  parts  of  the  other,  and  they  are  simi- 
larly arra^iged. 

For  the  triedrals  formed  by  the  given  faces  in  the 
two  must  be  equal,  and  may  therefore  coincide;  and 
the  given  faces  will  also  coincide,  being  equal.  But 
now  the  vertices  and  bases  of  the  two  pyramids  coin- 
cide. These  include  the  extremities  of  every  edge. 
Therefore,  the  edges  coincide ;  also  the  faces,  and  the 
figures  throughout. 

SIMILAR   PYRAMIDS. 

G55.  Theorem — Tivo  similar  pyramids  are  composed 
i>f  tetraedrons  respectively  similar,  ajid  similarly  arranged ; 
and,  conversely,  two  pyramids  are  similar  when  com^ 
posed  of  similar  tetraedrons,  similarly  arranged. 


224  ELEMENTS   OF   GEOMETRY. 

0>8.  Theorem — When  a  pyramid  is  cut  hy  a  plane 
parallel  to  the  base,  the  pyramid  cut  off  is  similar  to  the 
whole. 

These  theorems  may  be  demonstrated  by  the  student. 
Their  demonstration  is  like  that  of  analogous  proposi- 
tions in  triangles  and  tetraedrons. 


REGULAR    PYRAMIDS. 

657.  A  Regular  Pyramid  is  one  whose  base  is  a 
regular  polygon,  and  whose  vertex  is  in  the  line  perpen- 
dicular to  the  base  at  its  center. 

658.  Corollary. — The  lateral  edges  of  a  regular  pyra- 
mid are  all  equal  (529),  and  the  sides  are  equal  isosce- 
les triangles. 

659.  The  Slant  Hight  of  a  regular  pyramid  is  the 
perpendicular  let  fall  from  the  vertex  upon  one  side  of 
the  base.  It  is  therefore  the  altitude  of  one  of  the 
sides  of  the  pyramid. 

660.  Theorem. —  The  area  of  the  lateral  surface  of  a 
regular  pyramid  is  equal  to  half  the  product  of  the  pe- 
rimeter of  the  base  by  the  slant  hight. 

The  area  of  each  side  is  equal  to  half  the  product  of 
its  base  by  its  altitude  (386).  But  the  altitude  of  each 
of  the  sides  is  the  slant  hight  of  the  pyramid,  and  the 
su  n  of  all  the  bases  of  the  sides  is  the  perimeter  of  the 
base  of  the  pyramid. 

Therefore,  the  area  of  the  lateral  surface  of  the  pyr- 
amid, which  is  the  sum  of  all  the  sides,  is  equal  to  half 
the  product  of  the  perimeter  of  the  base  by  the  slant 
hight. 

661.  When  a  pyramid  is  cut  by  a  plane  parallel  to 
the  base,  that  part  of  the  figure  between  this  plane  and 


PYRAMIDS.  225 

the  base  is  called  a  frustum  of  a  pyramid,  or  a  trunc- 
ated pyramid. 

00!S.  Corollary — The  sides  of  a  frustum  of  a  pyra- 
mid are  trapezoids  (572) ;  and  the  sides  of  the  frustum 
of  a  regular  pyramid  are  equal  trapezoids. 

603.  The  section  made  by  the  cutting  plane  is  called 
the  upper  base  of  the  frustum.  The  slant  JdgJit  of  the 
frustum  of  a  regular  pyramid  is  that  part  of  the  slant 
hight  of  the  original  pyramid  which  lies  between  the 
bases  of  the  frustum.  It  is  therefore  the  altitude  of 
one  of  the  lateral  sides. 

664.  Theorem. — The  area  of  the  lateral  surface  of  the 
frustum  of  a  regular  pyramid  is  equal  to  half  the  prod- 
uct of  the  sum  of  the  perimeters  of  the  bases  by  the  slant 
hight. 

The  area  of  each  trapezoidal  side  is  equal  to  half  the 
product  of  the  sum  of  its  parallel  bases  by  its  altitude 
(392),  which  is  the  slant  hight  of  the  frustum.  There- 
fore, the  area  of  the  lateral  surface,  which  is  the  sum  of 
all  these  equal  trapezoids,  is  equal  to  the  product  of  half 
the  sum  of  the  perimeters  of  the  bases  of  the  frustum, 
multiplied  by  the  slant  hight. 

6®e5.  Corollary. — The  area  of  the  lateral  surface  of  a 
frustum  of  a  regular  pyramid  is  equal  to  the  product 
of  the  perimeter  of  a  section  midway  between  the  two 
bases,  multiplied  by  the  slant  hight.  For  the  perimeter 
of  a  section,  midway  between  the  two  bases,  is  equal  to 
half  the  sum  of  the  perimeters  of  the  bases. 

066.  Corollary — The  area  of  the  lateral  surface  of  a 
regular  pyramid  is  equal  to  the  product  of  the  slant 
hight  by  the  perimeter  of  a  section,  midway  between  the 
vertex  and  the  base.  For  the  perimeter  of  the  middle 
section  is  one-half  the  perimeter  of  the  base. 


226 


ELEMENTS    OF    GEOMETRY. 


MODEL    PYRAMIDS. 

GGlf*  The  student  may  construct  a  model  of  a  regular  pyra- 
mid. First,  draw  a  regular  polygon  of  any  number  of  sides. 
Upon  these  sides,  as  bases,  draw  equal  isosceles  triangles,  taking 
care  that  their  altitude  be  greater  than  the  apothem  of  the  base. 
The  figure  may  then  be  cut  out  and  folded. 


EXERCISES. 

668. — 1.  Find  the  area  of  the  surface  of  a  regular  octagonal 
pyramid  whose  slant  hight  is  5  inches,  and  a  side  of  whose  base 
IS  2  inches. 

2.  What  is  the  area  in  square  inches  of  the  entire  surface  of 
a  regular  tetraedron,  the  edge  being  one  inch  ?     Ans.    y/o. 

3.  A  pyramid  is  regular  when  its  sides  are  equal  isosceles 
triangles,  whose  bases  form  the  perimeter  of  the  base  of  the 
pyramid. 

4.  State  other  cases  of  equal  pyramids,  in  addition  to  those 
given.  Article  654. 

5.  When  two  pyramids  of  equal  altitude  have  their  bases  in 
the  same  plane,  and  are  cut  by  a  plane  parallel  to  their  bases, 
the  areas  of  the  sections  are  proportional  to  the  areas  of  the 
bases. 

PRIS3IS. 


669.  A  Prism  is  a  polyedron  which  has  two  of  its 
faces  equal  polygons  lying  in  par- 
allel planes,  and  the  other  faces 
parallelograms.  Its  possibility  is 
siown  by  supposing  two  equal  and 
pirallel  polygons  lying  in  two  par- 
allel planes  (569).  The  equal  sides 
being  parallel,  let  planes  unite  them. 
The  figure  thus  formed  on  each 
plane  is  a  parallelogram,  for  it  has 
two  opposite  sides  equal  and  parallel. 


rUlSMS.  227 

The  parallel  polygons  are  called  the  bases,  the  paral- 
lelograms the  sides  of  the  prism,  and  the  intersections 
of  the  sides  are  its  lateral  edges. 

The  altitude  of  a  prism  is  the  perpendicular  distance 
between  the  planes  of  its  bases. 

670.  Corollary. — The  lateral  edges  of  a  prism  are  all 
parallel    to    each    other,   and    therefore    equal    to    each 

other  (573). 

671.  A  Right  Prism  is  one  whose  lateral  edges  are 
perpendicular  to  the  bases. 

A  Regular  Prism  is  a  right  prism  whose  base  is  a 
regular  polygon. 

672.  Corollary — The  altitude  of  a  right  prism  is 
equal  to  one  of  its  lateral  edges ;  and  the  sides  of  a 
right  prism  are  rectangles.  The  sides  of  a  regular 
prism  are  equal. 

673.  Theorem — If  tivo  parallel  planes  pass  through 
a  prism,  so  that  each  j9?ang  cuis  every  lateral  edge,  the 
sections  made  by  the  two  planes  are  equal  polygons. 

Each  side  of  one  of  the  sections  is  parallel  to  the 
corresponding  side  of  the  other  section,  since  they  are 
the  intersections  of  two  parallel  planes  by  a  third. 
Hence,  that  portion  of  each  side  of  the  prism  which  is 
between  the  secant  planes,  is  a  parallelogram.  Since 
the  sections  have  their  sides  respectively  equal  and 
parallel,  their  angles  are  respectively  equal.  There- 
fore, the  polygons  are  equal. 

674.  Corollary — The  section  of  a  prism  made  by  a 
plane  parallel  to  the  base  is  equal  to  the  base,  and  the 
given  prism  is  divided  into  two  prisms.  If  two  paral- 
lel planes  cut  a  prism,  as  stated  in  the  above  theorem, 
that  part  of  the  solid  between  the  two  secant  planes  is 
also  a  prism. 


228 


ELEMENTS  OF  GEOMETRY. 


E 

f 

q:::;:.r>: 

71 

C 

,, 

^  ,L_..__j^ 

\ 

,,''' 

7 

HOW    DIVISIBLE. 

675.  Problem. — Every  prism  can  he  divided  into  the 
same  number  of  triangular  prisms  as  its  base  can  be  into 
triangles. 

If  homologous  diagonals  be  made  in  the  two  bases, 
as  EO  and  CF,  they  will  lie  in 
one  plane.  For  CE  and  OF 
being  parallel  to  each  other 
(670),  lie  in  one  plane.  There- 
fore, through  each  pair  of  these 
homologous  diagonals  a  plane 
may  pass,  and  these  diagonal 
planes  divide  the  prisms  into 
triangular  prisms. 

673.  Problem — A  triangular  prism  may  be  divided 
into  three  tetraedrons,  which,  taken  two  and  two,  have 
equal  bases  and  equal  altitudes. 

Let  a  diagonal  plane  pass  through  the  points  B,  C, 
and  II,  making  the  intersections 
BH  and  CII,  in  the  sides  DF  and 
DG.  This  plane  cuts  oiF  the  tet- 
raedron  BCDH,  which  has  for 
one  of  its  faces  the  base  BCD 
of  the  prism ;  for  a  second  face, 
the  triangle  BCH,  being  the  sec- 
tion made  by  the  diagonal  plane; 
and  for  its  other  two  faces,  the 
triangles  BDII  and  CDH,  each 
being  half  of  one  of  the  sides  of  the  prism. 

The  remainder  of  the  prism  is  a  quadrangular  pyra- 
mid, having  the  parallelogram  BCGF  for  its  base,  and 
H  for  its  vertex.  Let  it  be  cut  by  a  diagonal  plane 
through  the  points  IT,  G,  and  B. 


PllISMS.  229 

This  plane  separates  two  tetraedrons,  HBCG  and 
IIBFG.  The  two  faces,  HBC  and  HBG,  of  the  tetrae- 
dron  HBCGj  are  sections  made  by  the  diagonal  planes; 
and  the  two  faces,  HCG  and  BCG,  are  each  half  of  one 
side  of  the  prism.  The  tetraedron  HBFG  has  for  one 
of  its  faces  the  base  HFG  of  the  prism ;  for  a  second 
face,  the  triangle  HBG,  being  the  section  made  by  the 
diagonal  plane;  and,  for  the  other  two,  the  triangles 
HBF  and  GBF,  each  being  half  of  one  of  the  sides  of 
the  prism. 

Now,  consider  these  two  tetracdrons  as  having  their 
bases  BCG  and  BFG.  These  are  e((ual  triangles  lying 
in  one  plane.  The  point  H  is  the  common  vertex,  and 
therefore  they  have  the  same  altitude ;  that  is,  a  perpen- 
dicular from  H  to  the  plane  BCGF. 

Next,  consider  the  first  and  last  tetraedrons  described, 
HBCD  and  BFGH,  the  former  as  having  BCD  for  its 
base,  and  H  for  its  vertex ;  the  latter  as  having  FGH 
for  its  base,  and  B  for  its  vertex.  These  bases  are 
equal,  being  the  bases  of  the  given  prism.  The  vertex 
of  each  is  in  the  plane  of  the  base  of  the  other. 
Therefore,  the  altitudes  are  equal,  being  the  distance 
between  these  two  planes. 

Lastly,  consider  the  tetraedrons  BCDH  and  BCGH 
as  having  their  bases  CDH  and  CGH.  These  are  equal 
triangles  lying  in  one  plane.  The  tetraedrons  have  the 
common  vertex  B,  and  hence  have  the  same  altitude. 

677.  Corollary. — Any  prism  may  be  divided  into 
tetraedrons  in  several  ways ;  but  the  methods  above  ex- 
plained are  the  simplest. 

678.  Remark. — On  account  of  the  importance  of  the  above 
problem  in  future  demonstrations,  tlie  student  is  advised  to  make 
a  model  triangular  prism,  and  divide  it  into  tetraedrons.  A  po 
tato  may  be  used  for  this  purpose.  The  student  will  derive  most 
benefit  from  those  models  and  diagrams  which  he  nmkes  himself 


230  ELEMENTS    OF    GEOMETRY. 


EQUAL    PKISMS. 


OTO.  Theorem — Two  prisms  are  equal,  when  a  base 
and  two  adjacent  sides  of  the  one  are  respectively  equal  to 
the  corresponding  parts  of  the  other,  and  they  are  simi- 
larly arranged. 

For  the  triedrals  formed  by  the  given  faces  in  the 
two  prisms  must  be  equal  (599),  and  may  therefore  be 
made  to  coincide.  Then  the  given  faces  will  also  coin- 
cide, being  equal.  These  coincident  points  include  all 
of  one  base,  and  several  points  in  the  second.  But  the 
second  bases  have  their  sides  respectively  equal,  and 
parallel  to  those  of  the  first.  Therefore,  they  also  coin- 
cide, and  the  two  prisms  having  both  bases  coincident, 
must  coincide  throughout. 

G^O.  Corollary — Two  right  prisms  are  equal  v^licn 
they  have  equal  bases  and  the  same  altitude. 

GSl,  The  theory  of  similar  prisms  presents  nothing 
diiiicult  or  peculiar.  The  same  is  true  of  symmetrical 
prisms,  and  of  symmetrically  similar  prisms. 

AREA    OF    THE    SURFACE. 

O.S2.  Theorem. —  The  area  of  the  lateral  surface  of  a 
jirism  is  equal  to  the  p)rodiwt  of  one  of  the  lateral  edges 
hy  the  perimeter  of  a  section,  made  by  a  plane  per  pen- 
dijular  to  those  edges. 

Since  the  lateral  edges  are  parallel,  the  plane  TIN, 
parpendicular  to  one,  is  perpendicular  to  all  of  them. 
Therefore,  the  sides  of  the  polygon,  HK,  KL,  etc.,  are 
severally  perpendicular  to  the  edges  of  the  prism  which 
they  unite  (519). 

Then,  in  order  to  measure  the  area  of  each  face  of 
the  prism,  we  take  one  edge  of  the  prism  as  the  base 


PHISMS. 


231 


of  the  parallelogram,  and  one  side  of  the  polygon  HN 
as  its  altitude. 
Thus, 

area  AG  =  ABXHP, 

area  EB  =  EC  X  HK,  etc. 

By  addition,  the  sum  of  the 
areas  of  these  parallelograms  is 
the  lateral  surface  of  the  prism, 
and  the  sum  of  the  altitudes  of 
the  parallelograms  is  the  perim- 
eter of  the  polygon  HN.  Then, 
since  the  edges  are  equal,  the 
area  of  all  the  sides  is  equal  to 
the  product  of  one  edge,  multi- 
plieJ  by  the  perimeter  of  the 
polygon. 

083.  Corollary. — The  area  of  the  lateral  surface  of  a 
right  prism  is  equal  to  the  product  of  the  altitude  by 
the  perimeter  of  the  base. 

^84.  Corollary. — The  area  of  the  entire  surface  of  a 
regular  prism  is  equal  to  the  product  of  the  perime- 
ter of  the  base  by  the  sum  of  the  altitude  of  the  pris,m 
and  the  apothem  of  the  base. 


EXERCISES, 


6S5. — 1.  A  right  prism  has  less  surface  than  any  other 
prism  of  equal  base  and  equal  altitude;  and  a  regular  prism  lias 
less  surface  than  any  other  right  prism  of  equivalent  base  and 
equal  altitude. 

2.  A  regular  pyramid  and  a  regular  prism  have  equal  hexag- 
onal bases,  and  altitudes  equal  to  three  times  the  radius  of  the 
base;  required  the  ratio  of  the  areas  of  their  lateral  surfaces. 

3.  Demonstrate  the  principle  stated  in  Article  G83,  vvitliout  the 
aid  of  Article  682. 


232  ELEMENTS    OF   GEOMETRY. 

MEASURE    OF    VOLUME. 

086.  A  Parallelopiped  is  a  prism  whose  bases  are 
parallelograms.  Hence,  a  parallelopiped  is  a  solid  in- 
closed by  six  parallelograms. 

0^7.  Theorem. —  The  opposite  sides  of  a  parallelopiped 

are  equal. 

For  example,  the  faces  AI  and  BD  are  equal. 

For  10  and  DF  are  equal,  being  opposite  sides  of 
the  parallelogram  IF.  For 
a  like  reason,  EI  is  equal 
to  CD.  But,  since  these 
equal  sides  are  also  par- 
allel, the  included  angles 
EIO  and  CDF  are  equal. 
Hence,  the  parallelograms 
are  equal. 

68S.  Corollary — Any  two  opposite  faces  of  a  paral- 
lelopiped may  be  assumed  as  the  bases  of  the  figure. 

6^9.  A  parallelopiped  is  called  right  in  the  same 
case  as  any  other  prism.  When  the  bases  also  are 
rectangles,  it  is  called  rectangidar.  Then  all  the  faces 
are  rectangles. 

G90.  A  Cube  is  a  rectangular  parallelopiped  whose 
length,  breadth,  and  altitude  are  equal.  Then  a  cube 
is  a  solid,  bounded  by  six  equal  squares.  All  its  verti- 
ces, being  trirectangular  triedrals,  are  equal  (602).  All 
i.s  edges  are  of  right  diedral  angles,  and  therefore 
equal  (555). 

The  cube  has  the  simplest  form  of  all  geometrical 
solids.  It  holds  the  same  rank  among  them  that  the 
square  does  among  plane  figures,  ar.d  the  straight  line 
among  lines. 


MEASURE  OF  VOLUME. 


233 


The  cube  is  taken,  therefore,  as  the  unit  of  measure 
of  volume.  That  is,  Avhatever  straight  line  is  taken  as 
the  unit  of  length,  the  cube  whose  edge  is  of  that 
length  is  the  unit  of  volume,  as  the  square  whose  side 
is  of  that  length  is  the  measure  of  area. 


\  \  \  \  \ 

v\    \     \     \-^ 

1 

\    \    \    \ 

VOLUME     OF     PARALLELOPIPEDS. 

691.  Theorem. — The  volume  of  a  rectangular  paral- 
lelopiped  is  equal  to  the  product  of  its  length,  breadth, 
and  altitude. 

In  the  measure  of  the  rectangle,  the  product  of  one 
line  by  another  was  ex- 
plained. Here  we  have 
three  lines  used  with  a 
similar  meaning.  That 
is,  the  number  of  cu- 
bical units  contained  in 
a  rectangular  parallele- 
piped is  equal  to  the 
product  of  the  numbers 
of  linear  units  in  the  length,  the  breadth,  and  the  alti- 
tude. 

If  the  altitude  AE,  the  length  EI,  and  the  breadth 
10,  have  a  common  measure,  let  each  be  divided  by  it ; 
and  let  planes,  parallel  to  the  faces  of  the  prism,  pass 
through  all  the  points  of  division,  B,  C,  D,  etc. 

By  this  construction,  all  the  angles  formed  by  these 
planes  and  their  intersections  are  right  angles,  and  each 
of  the  intercepted  lines  is  equal  to  the  linear  unit  used 
in  dividing  the  edges  of  the  prism.  Therefore,  the 
prism  is  divided  into  equal  cubes.  The  number  of 
these  at  the  base  is  equal  to  the  number  of  rows,  mul- 
tiplied by  the  number  in  each  row;  that  is,  the  product 
Geom.— 20 


234  ELEMENTS    OF   GEOMETRY. 

of  the  length  by  the  breadth.  There  are  as  many 
layers  of  cubes  as  there  are  linear  units  of  altitude. 
Therefore,  the  whole  number  is  equal  to  the  product  of 
the  length,  breadth,  and  altitude.  In  the  diagram,  the 
dimensions  being  four,  three,  and  two,  the  volume  is 
twenty-four. 

But  if  the  length,  breadth,  and  altitude  have  no  com- 
mon  measure,  a  linear  unit  may  be  taken,  successively 
smaller  and  smaller.  In  this,  we  would  not  take  the 
whole  of  the  linear  dimensions,  nor  Avould  we  measure 
the  whole  of  the  prism.  But  the  remainder  of  botli 
would  grow  less  and  less.  The  part  of  the  prism  meas- 
ured at  each  step,  would  be  measured  exactly  by  the 
principle  just  demonstrated. 

By  these  successive  diminutions  of  the  unit,  we  can 
make  the  part  measured  approach  to  the  whole  prism  as 
nearly  as  we  please.  In  a  word,  the  whole  is  the  limit 
of  the  parts  measured ;  and  since  the  principle  demon- 
strated is  true  up  to  the  limit,  it  must  be  true  at  the 
limit.  Therefore,  the  rectangular  parallelepiped  is  meas- 
ured by  the  product  of  its  length,  breadth,  and  altitude. 

69!S.  Theorem — The  volume  of  any  parallelopiiyed  is 
equal  to  the  product  of  its  length,  hreadth,  and  altitude. 

Inasmuch  as  this  has  just  been  demonstrated  for  the 
rectangular  parallelepiped,  it  will  be  sufficient  to  show 
that  any  parallelepiped  is  equivalent  to  a  rectangular 
one  having  the  same  linear  dimensions. 

Suppose  the  lower  bases  of  the  two  prisms  to  be 
placed  on  the  same  plane.  Then  their  upper  bases  must 
also  be  in  one  plane,  since  they  have  the  same  altitude. 
Let  the  altitude  AE  be  divided  into  an  infinite  number 
of  equal  parts,  and  through  each  point  of  division  pass 
a  plane  parallel  to  the  base  AI. 

Now,  every  section  in   either  prism  is   equal   to   the 


MEASURE    OF    VOLUME. 


235 


base ;  but  the  bases  of  the  two  prisms,  having  the  same 
length  and  breadth,  are  equivalent.  The  several  par- 
tial infinitesimal  prisms  are  reduced  to  equivalent  fig 


ures.  Although  they  are  not,  strictly  speaking,  paral- 
lelograms, yet  their  altitudes  being  infinitesimal,  there 
can  be  no  error  in  considering  them  as  plane  figures ; 
which,  being  equal  to  their  respective  bases,  are  equiva- 
lent. Then,  the  number  of  these  is  the  same  in  each 
prism.  Therefore,  the  sum  of  the  whole,  in  one,  is 
equivalent  to  the  sum  of  the  whole,  in  the  other ;  that 
is,  the  two  parallelepipeds  are  equivalent. 

Besides  the  above  demonstration  by  the  method  of 
infinites,  the  theorem  may  be  demonstrated  by  the  or- 
dinary method  of  reasoning,  which  is  deduced  from 
principles  that  depend  upon  the  superposition  and  cc 
incidence  of  equal  figures,  as  follows ; 
Let  AF  be  any  oblique 

parallelopiped.    It  may  be 

shown  to  be  equivalent  to 

the      parallelopiped    AL, 

which   has   a  rectangular 

base,  AH,  since  the  prism 

LHEO    is    equal    to    the 

prism    DGAI.      But    the 

poTallelopipeds    AF     and 

AL  have  the  same  length,  breadth,  and  altitude. 


236 


ELEMENTS    OF    GEOMETRY. 


By  similar  reasoning,  the  prism  AL  may  be  shoAvn 
to  be  equivalent  to  a  prism  of  the  same  base  and  alti- 
tude, but  with  two  of  its  opposite  sides  rectangular. 
This  third  prism  may  then  be  shown  to  be  equivalent  to 
a  fourth,  which  is  rectangular,  and  has  the  same  dimen- 
sions as  the  others. 

69S.  Corollary. — The  volume  of  a  cube  is  equal  to 
the  third  power  of  its  edge.  Thence  comes  the  name  of 
cube,  to  designate  the  third  power  of  a  number. 

MODEL    CUBES. 


694.  Draw  six  equal  squares, 
as  in  the  diagram.  Cut  out  the 
fisure,  fold  at  the  dividing  lines,  and 
glue  the  edges.  It  is  well  to  have 
at  least  eight  of  one  size. 


CI05.  Corollary — The  volume  of  any  parallelopiped 
is  equal  to  the  product  of  its  base  by  its  altitude. 

696.  Corollary. — The  volumes  of  any  two  parallelo- 
pipeds  are  to  each  other  as  the  products  of  their  threj 
dimeusiond. 


VOLUME    OF    PRISMS. 

697.  Theorem. —  The  volume  of  any  triangular  pnf^m 
is  equal  to  the  product  of  its  base  by  its  altitude. 

The  base  of  any  right  triangular  prism  may  be  con- 
sidered as  one-half  of  the  base  of  a  right  parallelopiped. 
Then  the  whole  parallelopiped  is  double  the  given  prism, 
for  it  is  composed  of  two  right  prisms  having  equal 
bases  and  the  same  altitude,  of  which  the  given  prism 


MEASURE    OF    VOLUME. 


237 


is  one.  Therefore,  the  given  prism  is  measured  by  half 
the  product  of  its  altitude  by  the  base  of  the  parallel- 
epiped ;  that  is,  by  the  product  of  its  own  base  and 
altitude. 

If  the  given  prism  be  oblique,  it  may  be  shown,  by 
demonstrations  similar  to  the  first  of  those  in  Article 
692,  to  be  equivalent  to  a  right  prism  having  the  same 
base  and  altitude. 

698.  Corollary. — The  volume  of  any  prism  is  equal 
to  the  product  of  its  base  by  its  altitude.  For  any 
prism  is  composed  of  triangular  prisms,  having  the  com- 
mon altitude  of  the  given  prism,  and  the  sum  of  their 
bases  forming  the  given  base. 

699.  Corollary.  —  The  volume  of  a  triangular  prism 
is  equal  to  the  product  of  one  of  its  lateral  edges  mul- 
tiplied by  the  area  of  a  section  perpendicular  to  that 
edge. 


VOLUME    OF    TETRAEDRONS. 

700.  Theorem. —  Two   tetraedrons  of  equivalent  bases 
and  of  the  same  altitude  are  equivalent. 

Suppose  the  bases  of  the  two  tetraedrons  to  be  in  the 
E  C 


0  F 

same  plane.  Then  their  vertices  lie  in  a  plane  parallel 
to  the  bases,  since  the  altitudes  are  equal.  Let  the 
edge  AE  be  divided  into  an  infinite  number  of  parts, 


238  ELEMENTS   OF   GEOMETRY. 

and  through  each  point  of  division  pass  a  plane  parallel 
to  the  base  AIO.     - 

Now,  the  several  infinitesimal  frustums  into  which  the 
two  figures  are  divided  may,  without  error,  be  consid- 
ered as  plane  figures,  since  their  altitudes  are  infinitesi- 
mal. But  each  section  of  one  tetraedron  is  equivalent 
to  the  section  made  by  the  same  plane  in  the  other  tet- 
raedron. Therefore,  the  sum  o'f  all  the  infinitesimal 
frustums  in  the  one  figure  is  equivalent  to  the  sum  of 
all  in  the  other;  that  is,  the  two  tetraedrons  are  equiv- 
alent. 

701.  Theorem — The  volume  of  a  tetraedron  is  equal 
to  one-third  of  the  product  of  the  base  by  the  altitude. 

Upon  the  base  of  any  given  tetraedron,  a  triangular 
prism  may  be  erected,  which  shall  have  the  same  alti- 
tude, and  one  edge  coincident  with  an  edge  of  the  tet- 
raedron. This  prism  may  be  divided  into  three  tetrae- 
drons, the  given  one  and  two  others,  which,  taken  tAvo 
and  two,  have  equal  bases  and  altitudes  (676). 

Then,  these   three   tetraedrons   are  equivalent  (700); 
and  the  volume  of  the  given  tetraedron  is  one-third  of 
the  volume  of  the  prism;  that  is,  one-third  of  the  prod-, 
uct  of  its  base  by  its  altitude. 

VOLUME    OF    PYRAMIDS. 

702.  Corollary. — The  volume;  of  any  pyramid  is  equal 
to  one-third  of  the  product  of  its  base  by  its  altitude. 
For  any  pyramid  is  composed  of  triangular  pyramids; 
that  is,  of  tetraedrons  having  the  common  altitude  of 
the  given  pyramid,  and  the  sum  of  their  bases  forming 
the  given  base  (653). 

703.  Corollary. — The  volumes  of  two  prisms  of  equiv- 
alent bases  are  to  each  other  as  their  altitudes,  and  the 


SIMILAIl   POLYEDRONS.  229 

volumes  of  two  prisms  of  equal  altitudes  are  to  each 
other  as  their  bases.     The  same  is  true  of  pyramids, 

■704.  Corollary. — Symmetrical  prisms  are  equivalent. 
The  same  is  true  of  symmetrical  pyramids. 

705.  The  volume  of  a  frustum  of  a  pyramid  is  found 
by  subtracting  the  volume  of  the  pyramid  cut  off  from 
the  volume  of  the  whole.  When  the  altitude  of  the 
whole  is  not  given,  it  may  be  found  by  this  proportion  : 
the  area  of  the  lower  base  of  the  frustum  is  to  the  area, 
of  its  upper  base,  which  is  the  base  of  the  part  cut  off, 
as  the  square  of  the  whole  altitude  is  to  the  square  of 
the  altitude  of  the  part  cut  off. 

EXERCISES. 

tOG, — 1.  What  is  the  ratio  of  the  volumes  of  a  pyramid  and 
prism  having  the  same  base  and  altitude  ? 

2.  If  two  tetraedrons  have  a  triedral  vertex  in  each  equal, 
their  volumes  are  in  the  ratio  of  the  products  of  the  edges  wliich 
contain  the  equal  vertices. 

3.  The  plane  which  bisects  a  diedral  angle  of  a  tetraedron, 
divides  the  opposite  edge  in  the  ratio  of  the  areas  of  the  adjacent 
faces. 

SIMILAR    POLYEDRONS. 

TOT.  The  propositions  (640  to  643)  upon  the  ratios 
of  the  areas  of  the  surfaces  of  similar  tetraedrons,  may 
be  applied  by  the  student  to  any  similar  polyedrons. 
These  propositions  and  the  following  are  equally  appli- 
cable to  polyedrons  that  are  symmetrically  similar. 

TOS.  Problem. — An^  two  similar  polyedrons  may  he 
divided  into  the  same  number  of  similar  tetraedrons,  which 
shall  be  respectively  siinilar,  and  similarly  arranged. 

For,  after  dividing  one  into  tetraedrons,  the  construe 


240 


ELEMENTS   OF   GEOMETRY. 


tiori  of  the  homologous  lines  in  the  other  will  divide 
it  in  the  same  manner.  Then  the  similarity  of  the  re- 
spective tetraedrons  follows  from  the  proportionality  of 
the  lines. 

709.  Theorem. —  The  volumes  of  similar  poly edroyis  are 
proportional  to  the  cubes  of-  homologous  lines. 

First,  suppose  the  figures  to  be  tetraedrons.  Let 
AH  and  BG  be  the  altitudes. 


Then  (641),  EIO  :  CDF  : :  EF  :  CF^  ; :  AH^  :  BG^ 

)^y  the  proportionality  of  homologous  lines,  (634), 
J  AH  :  i  BG  :  :  EI  :  CF  : :  AH  :  BG. 

Multiplying  these  proportions  (701),  we  have 
AEIO  :  BCFD  : :  EF  :  CF'^  :  :  AH^  :  BG^ 
or,  as  the  cubes  of  any  other  homologous  lines. 

Next,  let  any  two  similar  polyedrons  be  divided  into 
the  same  number  of  tetraedrons.  Then,  as  just  proved, 
the  volumes  of  the  homologous  parts  are  proportional  to 
the  cubes  of  the  homologous  lines.  By  arranging  these 
in  a  continued  proportion,  as  in  Article  436,  we  may 
show  that  the  volume  of  either  polyedron  is  to  the  vol- 
ume of  the  other  as  the  cube  of  any  line  of  the  first  is 
to  the  cube  of  the  homologous  line  of  the  second. 


REGULAR    POLYEDRONS. 


241 


710.  Notice  that  in  the  measure  of  every  area  there 
are  two  linear  dimensions ;  and  in  the  measure  of  every 
volume,  three  linear,  or  one  linear  and  one  superficiaL 

• 

EXERCISE. 

Ill,  What  is  the  ratio  between  the  edges  of  two  cubes,  one  of 
wliich  lias  twice  the  volume  of  the  other? 

This  problem  of  the  duplication  of  the  cube  was  one  of  the 
celebrated  problems  of  ancient  times.  It  is  said  that  the  oracle 
of  Apollo  at  Delphos,  demanded  of  the  Athenians  a  new  altar, 
of  the  same  shape,  but  of  twice  the  volume  of  the  old  one.  The 
efforts  of  the  Greek  geometers  were  chiefly  aimed  at  a  graphic  so- 
lution; that  is,  the  edge  of  one  cube  being  given,  to  draw  a  line 
equal  to  the  edge  of  the  other,  using  no  instruments  but  the  rule 
and  compasses.  In  this  they  failed.  The  student  will  find  no 
difficulty  in  making  an  arithmetical  solution,  within  any  desirec? 
degree  of  approximation. 

REGULAR    POLYEDRONS. 

712.  A  Regular  Polyedron  is  one  whose  faces  arc- 
equal  and  regular  polygons,  and  whose  vertices  are  equal 
polyedrals. 


The  regular  tetraedron  and  the  cube,  or  regular  hexa- 
edron,  have  been  described. 

The  regular  ocfaedron  has  eight,  the  dodecaedron 
twelve,  and  the  icosaedron  twenty  faces. 


Geom.— 21 


£42  ELEMENTS   OF   GEOMETRY. 

The  class  of  figures  here  defined  must  not  be  con- 
founded with  regular  pyramids  or  prisms. 

713.  Problem. — It  is  not  possible  to  make  more  than 
five  regular  polyedrons.  ' 

First,  consider  thos3  whose  faces  are  triangles.  Each 
angle?  of  a  regular  triangle  is  one-third  of  two  right 
angles.  Either  three,  four,  or  five  of  these  may  bo 
joined  to  form  one  polyedral  vertex,  the  sum  being,  in 
each  case,  less  than  four  right  angles  (612).  But  the 
sum  of  six  such  angles  is  not  less  than  four  right 
angles.  Therefore,  there  can  not  be  more  than  three 
kinds  of  regular  polyedrons  whose  faces  are  triangles, 
viz. :  the  tetraedron,  where  three  plane  angles  form  a 
vertex ;  the  octaedron,  where  four,  and  the  icosaedron, 
where  five  angles  form  a  vertex. 

The  same  kind  of  reasoning  shows  that  only  one 
regular  polyedron  is  possible  with  square  faces,  the 
cube ;  and  only  one  with  pentagonal  faces,  the  dode- 
caedron. 

Regular  hexagons  can  not  form  the  faces  of  a  regular 
polyedron,  for  three  of  the  angles  of  a  regular  hexagon 
are  together  not  less  than  four  right  angles ;  and  there- 
fore they  can  not  form  a  vertex. 

So  much  the  more,  if  the  polygon  has  a  greater  num- 
ber of  sides,  it  will  be  impossible  for  its  angles  to  be 
the  faces  of  a  polyedral.  Therefore,  no  polyedron  is 
possible,  except  the  five  that  have  been  described. 

MODEL  KEGULATt  POLYEDKONS. 

')'14.  The  possibility  of  regular  polyedrons  of  eight,  of  twelve, 
and  of  twenty  sides  is  here  assumed,  as  the  demonstration  would 
occupy  more  space  than  the  principle  is  worth.  However,  the 
student  may  construct  models  of  these  as  follows.  Plans  for  ih*? 
regular  tetraedron  and  the  cube  have  already  been  given. 


REGULAR   POLYEDRONS. 


213 


For  the  octaedron,  draw 
eight  equal  regular  trian- 
gles, as  ill  the  diagram. 


For  the  dodecaedron,  draw 
twelve  equal  regular  penta- 
gons, as  in  the  diagram. 


For  the  icosacdron,  draw 
twenty  equal  regular  trian- 
gles, as  in  the  diagram. 


There   are  many  crystals,  which,  though    not   regular,  in    iIk 
geometrical  rigor  of  the  word,  yet  present  a  certain  regularity  of 


shape. 


EXERCISES. 


•715.— 1.  How  many  edges  and  how  many  vertices  has  each 
of  the  regular  polyedrons? 

2.  Calling  that  point  the  center  of  a  triangle  which  is  the  inter- 
section of  straight  lines  from  each  vertex  to  the  center  of  the 
opposite  side;  then,  demonstrate  that  the  four  lines  wliich  join  the 
vertices  of  a  tetraedron  to  the  centers  of  the  opposite  faces,  inter- 
sect each  other  in  one  point. 

3.  In  what  ratio  do  the  lines  just  described  in  the  tetraedron 
divide  each  other? 

4.  The  opposite  vertices  of  a  parallelepiped  are  symmetrical 
triedrals. 

5  The  diagonals  of  a  parallelopiped  bisect  each  other ;  the 
lines  which  join  the  centers  of  the  opposite  edges  bisect  each 
other;   the  lines  which  join  the  centers  of  tlie  opposite  faces  bi- 


244  ELEMENTS    OF   GEOMETRY. 

sect  each  o^htr;  and  the  point  of  intersection  is  the  same  for  all 
these  lines. 

6.  The  diagonals  of  a  rectangular  parallelopiped  are  equal, 

7.  The  square  of  the  diagonal  of  a  rectangular  parallelopiped 
is  equivalent  to  the  sum  of  the  squares  of  its  length,  breadth,  and 
altitude, 

8.  A  cube  is  the  largest  parallelopiped  of  the  same  extent  of 
surface, 

9.  It'  a  right  prism  is  symmetrical  to  another,  they  are  equal. 

10.  Within  any  regular  polyedron  there  is  a  point  equally 
distant  from  all  the  faces,  and  also  from  all  the  vertices. 

11.  Two  regular  polyedrons  of  the  same  number  of  faces  are 
similar. 

12.  Any  regular  polyedron  may  be  divided  into  as  many  regu- 
lar and  equal  pyramids  as  it  has  faces. 

13.  Two  different  tetraedrons,  and  only  two,  may  be  formed 
with  the  same  four  triangular  faces;  and  these  two  tetraedrons 
are  symmetrical. 

14.  The  area  of  the  lower  base  of  a  frustum  of  a  pyramid  is 
f^^e  square  feet,  of  the  upper  base  one  and  four-fifths  square  feet, 
and  the  altitude  iu  two  feet;  required  the  volume. 


SOLIDS    OF    REVOLUTION. 


245 


CHAPTER    XI 


SOLIDS   OF   REVOLUTION 


710.  Of  the  infinite  variety  of  forms  there  remain 
but  three  to  be  considered  in  this  elementary  work. 
These  are  formed  or  generated  by  the  revolution  of  a 
plane  figure  about  one  of  its  lines  as  an  axis.  Figures 
formed  in  this  way  are  called  solids  of  revolution. 

717.  A  Cone  is  a  solid  formed  by  the  revolution  of 
a  right  angled  triangle  about  one  of  its 
legs  as  an  axis.  The  other  leg  revolv- 
ing describes  a  plane  surface  (521). 
This  surface  is  also  a  circle,  having  for 
its  radius  the  leg  by  which  it  is  de- 
scribed. The  hypotenuse  describes  a 
curved  surface. 

The  plane  surface  of  a  cone  is  called  its  base.  The 
opposite  extremity  of  the  axis  is  the  vertex.  The  alti- 
tude is  the  distance  from  the  vertex  to  the  base,  and  the 
slant  Mglit  is  the  distance  from  the  vertex  to  the  cir- 
cumference of  the  base. 

718.  A  Cylinder  is  a  solid  described 
by  the  revolution  of  a  rectangle  about 
one  of  its  sides  as  an  axis.  As  in  the 
cone,  the  sides  adjacent  to  the  axis  de- 
scribe circles,  while  the  opposite  side 
describes  a  curved  surftice. 

The  plane  surfaces  of  a  cylinder  are  called  its  bases ^ 


246  ELEMENTS    OF   GEOMETllY. 

and  the  perpendicular  distance  between  them  is  its 
altitude. 

These  figures  are  strictly  a  regular  cone  and  a  regular 
cylinder,  yet  but  one  word  is  used  to  denote  the  figures 
defined,  since  other  cones  and  cylinders  are  not  usually 
discussed  in  Elementary  Geometry.  The  sphere,  which 
is  described  by  the  revolution  of  a  semicircle  about  the 
diameter,  will  be  considered  separately. 

TIO,  As  the  curved  surfaces  of  the  cone  and  of  the 
cylinder  are  generated  by  the  motion  of  a  straight  line, 
it  follows  that  each  of  these  surfaces  is  straight  in  one 
direction. 

A  straight  line  from  the  vertex  of  the  cone  to  the 
circumference  of  the  base,  must  lie  wholly  in  the  sur- 
face. So  a  straight  line,  perpendicular  to  the  base  of  a 
cylinder  at  its  circumference,  must  lie  wholly  in  the 
surface.  For,  in  each  case,  these  positions  had  been 
occupied  by  the  generating  lines. 

One  surface  is  tangent  to  another  w^hen  it  meets,  but 
being  produced  does  not  cut  it.  The  place  of  contact 
of  a  plane  with  a  conical  or  cylindrical  surface,  must 
be  a  straight  line ;  since,  from  any  point  of  one  of  those 
surfaces,  it  is  straight  in  one  direction. 

CONIC    SECTIONS. 

•720.  Every  point  of  the  line  which  describes  the 
curved  surface  of  a  cone,  or  of  a  cylinder,  moves  in  a 
plane  parallel  to  the  base  (565).  Therefore,  if  a  cone 
or  a  cylinder  be  cut  by  a  plane  parallel  to  the  base,  the 
section  is  a  circle. 

If  we  conceive  a  cone  to  be  cut  by  a  plane,  the  curve 
formed  by  the  intersection  will  be  diff'erent  according  to 
the  position  of  the  cutting  plane.     There  are  three  dif- 


CONES.  247 

ferent  modes  in  which  it  is  possible  for  the  intersection 
to  take  place.  The  curves  thus  formed  are  the  ellipse, 
parabola,  and  hyperbola. 

These  Conic  Sections  are  not  usually  considered  in 
Elementary  Geometry,  as  their  properties  can  be  better 
investigated  by  the  application  of  algebra. 

CONES. 

•ySl.  A  cone  is  said  to  be  inscribed  in  a  pyramid, 
when  their  bases  lie  in  one  plane,  and  the  sides  of  the 
pyramid  are  tangent  to  the  curved  surface  of  the  cone. 
The  pyramid  is  said  to  be  circumscribed  about  the  cone. 

A  cone  is  said  to  be  circumscribed  about  a  pyramid, 
when  their  bases  lie  in  one  plane,  and  the  lateral  edges 
of  the  pyramid  lie  in  the  curved  surface  of  the  cone. 
Then  the  pyramid  is  inscribed  in  the  cone. 

73!S.  Theorem — A  cone  is  the  limit  of  the  pyramids 
which  can  be  circumscribed  about  it;  also  of  the  pyramids 
which  can  be  inscribed  in  it. 

Let  ABODE  be  any  pyramid  circumscribed  about  a 
cone. 

The  base  of  the  cone  is  a 
circle  inscribed  in  the  base 
of  the  pyramid.  The  sides 
of  the  pyramid  are  tangent 
to  the  surface  of  the  cone. 

Now,  about  the  base  of  the 
cone  there  may  be  described 
a  polygon  of  double  the  num- 
ber of  sides  of  the  first,  each 

alternate  side  of  the  second  polygon  coinciding  with  a 
side  of  the  first.  This  second  polygon  may  be  the  base 
of  a  pyramid,  having  its  vertex  at  A.  Since  the  sides 
of  its  bases  are  tangent  to  the  base  of  the  cone,  every 


248  ELEMENTS    OF    GEOMETRY. 

side  of  the  pyramid  is  tangent  to  the  curved  surface  of 
the  cone.  Thus  the  second  pyramid  is  circumscribed 
about  the  cone,  but  is  itself  within  the  first  pyramid. 

By  increasing  the  number  of  sides  of  the  pyramid,  it 
can  be  made  to  approximate  to  the  cone  within  less 
than  any  appreciable  difference.  Then,  as  the  base  of 
the  cone  is  the  limit  of  the  bases  of  the  pyramids,  the 
cone  itself  is  also  the  limit  of  the  pyramids. 

Again,  let  a  polygon  be  inscribed  in  the  base  of  the 
cone.  Then,  straight  lines  joining  its  vertices  with  the 
vertex  of  the  cone  form  the  lateral  edges  of  an  inscribed 
pyramid.  The  number  of  sides  of  the  base  of  the  pyr- 
amid, and  of  the  pyramid  also,  may  be  increased  at 
will.  It  is  evident,  therefore,  that  the  cone  is  the 
limit  of  pyramids,  either  circumscribed   or  inscribed. 

723.  Corollary. — The  area  of  the  curved  surface  of 
a  cone  is  equal  to  one-half  the  product  of  the  slant  hight 
by  the  circumference  of  the  base  (660).  Also,  it  is 
equal  to  the  product  of  the  slant  hight  by  the  circumfer- 
ence of  a  section  midway  between  the  vertex  and  the 
base  (Q66). 

724.  Corollary. — The  area  of  the  entire  surface  of  a 
cone  is  equal  to  half  of  the  product  of  the  circumfer^ 
ence  of  the  base  by  the  sum  of  the  slant  hight  and  the 
radius  of  the  base  (499). 

725.  Corollary — The  volume  of  a  cone  is  equal  to 
one-third  of  the  product  of  the  base  by  the  altitude. 

726.  The  frustum  of  a  cone  is  defined  in  the  same 
way  as  the  frustum  of  a  pyramid. 

727.  Corollary — The  area  of  the  curved  surface  of 
the  frustum  of  a  cone  is  equal  to  half  the  product  of  its 
slant  hight  by  the  sum  of  the  circumferences  of  its  bases 
(664).     Also,  it  is   equal   to   the   product   of  its    slant 


CYLINDERS. 


249 


higlit  by  the  circumference  of  a  section  midway  between 
the  two  bases  (665). 

TSS.  Corollary. — If  a  cone  be  cut  by  a  plane  paral- 
lel to  the  base,  the  cone  cut  off  is  similar  to  the  whole 
(656). 

EXERCISES. 

'7^9, — 1.  Two  cones  are  similar  when  they  are  generated  by 
similar  triangles,  homologous  sides  being  used  for  the  axes. 

2.  A  section  of  a  cone  by  a  plane  passing  through  the  vertex, 
is  an  isosceles  triangle. 


^ 


^ 


CYLINDERS. 

TSO.  A  cylinder  is  said  to  be  in- 
scribed in  a  prism,  when  their  bases 
lie  in  the  same  planes,  and  the  sides 
of  the  prism  are  tangent  to  the  curved 
surface  of  the  cylinder.  The  prism  is 
then  said  to  be  circumscribed  about 
the  cylinder. 

A  cylinder  is  said  to  be  circum- 
scribed about  a  prism,  when  their  bases 
lie  in  the  same  planes,  and  the  lat- 
eral edges  of  the  prism  lie  in  the 
curved  surface  of  the  cylinder ;  and 
the  prism  is  then  said  to  be  inscribed 
in  the  cylinder. 

731.  Theorem. — A  cylinder  is  the  limit  of  the  prisms 
which  can  be  circumscribed  about  it;  also  of  those  which 
can  be  inscribed  in  it. 

The  demonstration  of  this  theorem  is  so  similar  to 
that  of  the  last,  that  it  need  not  be  repeated. 


250 


ELEMENTS   OF   GEOMETRY. 


732.  Corollary.— The  area  of  the  curved  surface  of  a 
cylinder  is  equal  to  the  product  of  the  altitude  by  the 
circumference  of  the  base  (683). 

733.  Corollary — The  area  of  the  entire  surface  of  a 
cylinder  is  equal  to  the  product  of  the  circumference  of 
the  base  by  the  sum  of  the  altitude  and  the  radius  of 
the  base  (684). 

734.  Corollary. — The  volume  of  a  cylinder  is  equal 
to  the  product  of  the  base  by  the  altitude  (698). 

MODEL    CONES    AND    CYLINDERS. 

T35.  Models  of  cones  and  cylinders  may  be  made  from  paper, 
taking  a  sector  of  a  circle  for  the  curved  surface  of  a  cone,  and 
a  rectangle  for  the  curved  surface  of  a  cylinder.  Make  the  bases 
separately. 

EXERCISES. 

'7SG, — 1.  Apply  to  cones  and  cylinders  the  principles  demon- 
strated of  similar  polyedrons. 

2.  A  section  of  a  cylinder  made  by  a  plane  perpendicular  to  the 
base  is  a  rectangle. 

3.  The  axis  of  a  cone  or  of  a  cylinder  is  equal  to  its  altitude. 

SPHERES. 


737.  A  Sphere  is  a  solid  de- 
scribed by  the  revolution  of  a 
semicircle  about  its  diameter  as 
an  axis. 

The  eenter,  radius,  and  diame- 
ier  of  the  sphere  are  the  same 
as  those  of  the  generating  circle. 
The  spherical  surface  is  described  by  the  circumference. 


SPHERES.  251 

•ySS.  Corollary — Every  point  on  the  surface  of  the 
sphere  is  equally  distant  from  the  center. 

This  property  of  the  sphere  is  frequently  given  as  its 
definition. 

T39.  Corollary. — All  radii  of  the  same  sphere  are 
equal.     The  same  is  true  of  the  diameters. 

•740.  Corollary. — Spheres  having  equal  radii  are  equal. 

741.  Corollary — A  plane  passing  through  the  center 
of  a  sphere  divides  it  into  equal  parts.  The  halves  of 
a  sphere  are  called  hemispheres. 

•742.  Theorem. — A  plane  which  is  perpendicular  to  a 
radius  of  a  sphere  at  its  extremity  is  tangent  to  the  sphere- 

For  if  straight  lines  extend  from 
the   center   of   the   sphere   to   any        ^^p^—       -  -=-^ 
other  point  of  the  plane,  they  are      ^^' 
oblique  and  longer  than  the  radius,        ^fc-  1/^^^ 
which  is  perpendicular  (530).  There-        ^^fc^^W 
fore,  every  point  of  the  plane  except  ^B^^^^ 

one  is    beyond   the  surface  of  the 
sphere,  and  the  plane  is  tangent. 

743.  Corollary — The  spherical  surface  is  curved  in 
every  direction.  Unlike  those  surfaces  which  are  gen- 
erated by  the  motion  of  a  straight  line,  every  possible 
section  of  it  is  a  curve. 


SECANT    PLANES. 

744.  Theorem.—  Every  section  of  a  sphere  made  by  a 
plane  is  a  circle. 

If  the  plane  pass  through  the  center  of  the  sphere, 
every  point  in  the  perimeter  of  the  section  is  equally 
distant  from  the  center,  and  therefore  the  section  is  a 
circle. 


2r)2  ELEMENTS    OF    GEOMETRY 

But  if  the  section  do  not  pass  through  the  center,  as 
DGF,  then  from  the  center  C  let  CI  fall  perpendicu- 
larly on  the  cutting  plane. 
Let  radii  of  the  sphere,  as 
CD  and  CG,  extend  to  diifer- 
ent  points  of  the  boundary 
of  the  section,  and  join  ID 
and  IG. 

Now  the  oblique  lines  CD 
and  CG  being  equal,  the 
points  D  and  G  must  be 
equally  distant  from  I,  the  foot  of  the  perpendicular 
(529).  The  same  is  true  of  all  the  points  of  the  pe- 
rimeter DGF.  Therefore,  DGF  is  the  circumference  of 
a  circle  of  which  I  is  the  center. 

745.  Corollary. — The  circle  formed  by  the  section 
through  the  center  is  larger  than  one  formed  by  any 
plane  not  through  the  center.  For  the  radius  BC  is 
equal  to  GO,  and  longer  than  GI  (104). 

746.  When  the  plane  passes  through  the  center  of  a 
sphere,  the  section  is  called  a  great  circle;  otherwise  it 
is  called  a  small  circle. 

747.  Corollary — All  great  circles  of  the  same  sphere 
are  equal. 

748.  Corollary. — Two  great  circles  bisect  each  other, 
and  their  intersection  is  a  diameter  of  the  sphere. 

749.  Corollary — If  a  perpendicular  be  let  fall  from 
the  center  of  a  sphere  on  the  plane  of  a  small  circle, 
the  foot  of  the  perpendicular  is  the  center  of  the  cir- 
cle ;  and  conversely,  the  axis  of  any  circle  is  a  diame- 
ter of  the  sphere. 

The  two  points  where  the  axis  of  a  circle  pierces  the 
spherical   surface,  are   the  poles  of  the   circle.     Thus, 


SPHERES.  253 

JS  and  S  are  the  poles  of  both  the  sections  in  the  last 
diagram. 

TSO.  Corollary. — Circles  whose  planes  are  parallel  to 
each  other  have  the  same  axis  and  the  same  poles. 


ARC    OF    A    GREAT    CIRCLE. 

751.  Theorem. —  The  shortest  line  which  can  extend 
from  one  point  to  another  along  (he  surface  of  a  sphere, 
is  the  arc  of  a  great  circle,  passing  through  the  ttvo  poiyits. 

Only  one  great  circle  can  pass  through  two  given 
points  on  the  surface  of  a  sphere  ;  for  these  two  points 
and  the  center  determine  the  position  of  the  plane  of 
the  circle. 

Let  ABCDEFG  be  any  curve  whatever  on  the  sur- 
face of  a  sphere  from  G 
to  A.  Let  AKG  be  the  arc 
of  a  great  circle  joining 
these  points,  and  also  AD 
and  DG  arcs  of  great  cir- 
cles joining  those  points 
with  the  point  D  of  the  given  curve. 

Then  the  sum  of  AD  and  DG  is  greater  than  AKG. 

For  the  planes  of  these  arcs  form  a  triedral  whose 
vertex  is  at  the  center  of  the  sphere.  These  arcs  have 
the  same  ratios  to  each  other  as  the  plane  angles  which 
compose  this  triedral,  for  the  arcs  are  intercepted  by 
the  sides  of  the  angles,  and  they  have  the  same  radius. 
But  any  one  of  these  angles  is  less  than  the  sum  of 
the  other  two  (586).  Therefore,  any  one  of  the  arcs  is 
less  than  the  sum  of  the  other  two. 

Again,  let  AH  and  HD  be  arcs  of  great  circles  join- 
ing A  and  D  with  some  point  H  of  the  given  curve ; 
also  let  DI  and   IG  be   arcs  of  great   circles.     In   the 


254  ELEMENTS   OF    GEOMETRY. 

same  manner  as  above,  it  may  be  shown  that  AH  and 
HD  are  greater  than  AD,  and  that  the  sum  of  DI  and 
IG  is  greater  than  DG.  Therefore,  the  sum  of  AH,  HD^ 
DI,  and  IG  is  still  greater  than  AKG. 

By  continuing  to  take  intermediate  points  and  join- 
ing them  to  the  preceding,  a  series  of  lines  is  formed, 
each  greater  than  the  preceding,  and  each  approaching 
nearer  to  the  given  curve.  Evidently,  this  approach  can 
be  made  as  nearly  as  we  choose.  Therefore,  the  curve 
is  the  limit  of  these  lines,  and  partakes  of  their  common 
character,  in  being  greater  than  the  arc  of  a  great  circle 
Avhich  joins  its  extremities. 

75^.  Theorem — Every  plane  passing  through  the  axis 
of  a  circle  is  'perpendicular  to  the  plane  of  that  circle,  and 
its  section  is  a  great  circle. 

The  first  part  of  this  theorem  is  a  corollary  of  Arti- 
cle 556.  The  second  part  is  proved  by  the  fact  that 
every  axis  passes  through  the  center  of  a  sphere  (749). 

753.  CDrollary — The  distances  on  the  spherical  sur- 
face from  any  points  of  a  circumference  to  its  pole,  are 
the  same.  For  the  arcs  of  great  circles  which  mark 
these  distances  are  equal,  since  all  their  chords  are 
equal  oblique  lines  (529). 

754.  Corollary. — The  distance  of  the  pole  of  a  great 
circle  from  any  point  of  the  circumference  is  a  quad- 
rant. 

APPLICATIONS. 

•755.  The  student  of  geography  will  recognize  the  equator  as 
a  great  circle  of  tlie  earth,  which  is  nearly  a  sphere.  The  paral- 
lels of  latitude  are  small  circles,  all  having  the  same  poles  as  the 
equator.  The  meridians  are  great  circles  perpendicular  to  the 
equator. 

The   application  of  the  principle  of  Article  751  to  navigation 


SPHERES. 


255 


has  been  one  of  the  greatest  reforms  in  tliat  art.  A  vessel  cross- 
ing liie  ocean  Ironi  a  port  in  a  certain  latitude  to  a  port  in  the 
same  latitude,  should  not  sail  along  a  parallel  of  latitude,  for  that 
is  the  arc  of  a  small  circle. 

T56.  The  curvature  of  tlie  sphere  in  every  direction,  renders 
it  impossible  to  construct  an  exact  model  with  plane  paper.  But 
the  student  is  advised  to  procure  or  make  a  globe,  upoi\  which  he 
can  draw  the  diagrams  of  all  the  figures.  This  is  the  more  im- 
portant on  account  of  the  difficulty  of  clearly  representing  these 
figures  by  diagrams  on  a  plane  surface. 


SPHERICAL    ANGLES. 


757.  A  Spherical  Angle  is  the  difference  in  the 
directions  of  two  arcs  of  great  cir- 
cles at  their  point  of  meeting.  To 
obtain  a  more  exact  idea  of  this 
angle,  notice  that  the  direction  of 
an  arc  at  a  given  point  is  the  same 
as  the  direction  of  a  straight  line 
tangent  to  the  arc  at  that  point. 
Thus,  the  direction  of  the  arc  BDF 
at  the  point  B,  is  the  same  as  the 
direction  of  the  tangent  BH. 

75S.  Corollary. — A  spherical  angle  is  the  same  as 
the  plane  angle  formed  by  lines  tangent  to  the  given 
arcs  at  their  point  of  meeting.  Thus,  the  spherical 
angle  DBG  is  the  same  as  the  plane  angle  HBK,  the 
lines  HB  and  BK  being  severally  tangent  to  the  arcs 
BD  and  BG. 

759.  Corollary. — A  spherical  angle  is  the  same  as 
the  diedral  angle  formed  by  the  planes  of  the  two  arcs. 
For,  since  the  intersection  BF  of  the  planes  of  the  arcs 
is  a  diameter  (748),  the  tangents  HB  and  KB  are  both 
perpendicular  to  it,  and  their  angle  measures  the  diedral. 


256 


ELEMENTS    OF   GEOMETRY. 


7S0.  Corollary. — A  spherical  an- 
gle is  measured  by  the  arc  of  a  cir- 
cle included  between  the  sides  of 
the  angle,  the  pole  of  the  arc  being 
at  the  vertex. 

Thus,  if  DG  is  an  arc  of  a  great 
circle  whose  pole  is  at  B,  then  the 
spherical  angle  DBG  is  measured 
by  the  arc  DG. 

T61«  A  LuNE  is  that  portion  of  the  surface  of  a 
sphere  included  between  two  halves  of  great  circles. 

That  portion  of  the  sphere  included  between  the  two 
planes  is  called  a  spherical  wedge.  Hence,  two  great 
circles  divide  the  surface  into  four  lunes,  and  the  sphere 
into  four  wedges. 


SPHERICAL    POLYGONS. 

76ti.  A  Spherical  Polygon  is  that  portion  of  the 
surface  of  a  sphere  included  between  three  or  more 
arcs  of  great  circles. 

Let  C  be  the  center  of  a  sphere,  and  also  the  vertex 
of  a  convex  polyedral.  Then, 
the  planes  of  the  faces  of  this 
polyedral  will  cut  the  surface 
of  the  sphere  in  arcs  of  great 
circles,  which  form  the  poly- 
gon BDFGH.  We  say  con- 
vex^ for  only  those  polygons 
which  have  all  the  angles 
convex   are   considered   among 

spherical  polygons.  Conversely,  if  a  spherical  polygon 
ha,ve  the  planes  of  its  several  sides  produced,  they  form 
a  polyedral  whose  vertex  is  at  the  center  of  the  sphere. 


SPHERES.  257 

The  angles  of  the  polygon  are  the  same  as  the  die- 
dral  angles  of  the  polyedral  (759). 

763.  Theorem. —  The  sum  of  all  the  sides  of  a  spher- 
ical polygon  is  less  than  a  circumference  of  a  great  circle. 

The  arcs  which  form  the  sides  of  the  polygon  measure 
the  angles  which  form  the  faces  of  the  corresponding 
polyedral,  for  all  the  arcs  have  the  same  radius. 

But  the  sum  of  all  the  faces  of  the  polyedral  being 
less  than  four  right  angles,  the  sum  of  the  sides  must 
be  less  than  a  circumference. 

761.  Theorem. — A  spherical  polygon  is  always  within 
the  surface  of  a  hemisphere. 

For  a  plane  may  pass  through  the  vertex  of  the  cor- 
responding polyedral,  having 
all  of  the  polyedral  on  one  side 
of  it  (609).  The  section  formed 
by  this  plane  produced  is  a 
great  circle,  as  KLM.  But 
since  the  polyedral  is  on  one 
side  of  this  plane,  the  corres- 
ponding polygon  must  be  con- 
tained within  the  surface  on 
one  side  of  it. 

7S5.  That  portion  of  a  sphere  which  is  included  be- 
tween a  spherical  polygon  and  its  corresponding  polye- 
dral is  called  a  spherical  pyramid,  the  polygon  being  its 
base. 

SPHERICAL   TRIANGLES. 

708.  If  the  three  planes  which  form  a  triedral  at 
the  center  of  a  sphere  be  produced,  they  divide  the 
sphere  into  eight  parts  or  spherical  pyramids,  each  hav- 
ing its  triedral  at  the  center,  and  its  spherical  triangle 

Geom.— 22 


•^58  ELEMENTS    OF    GEOMETRY. 

at  the  surface.     Thus,  for  every  spherical  triangle,  there 
are    seven   others    whose    sides    are    respectively    either 
equal  or  supplementary  to  those 
of  the  given  triangle.  y^      ^^^^^^>\ 

Of  these  seven  spherical  tri-         /         F/     /    ;\ 
angles,  that  which  lies  vertically       \C^^--^J  "^  /^^/      \ 
opposite  the  given  triangle,  as         \   /     7^--,^     \ 
GKH    to    FDB,   has   its    sides       \      /    7-.. '>  /  ""'/b 
respectively  equal  to  the  sides  \  I  /     /^      "y 

of  the  given  triangle,  but  they  g'^<l      ^^^ 

are  arranged  in  reverse  order ; 

for  the  corresponding  triedrals  are  symmetrical.     Such 
spherical  triangles  are  called  symmetrical. 

*767.  Corollary. — If  two  spherical  triangles  are  equal, 
their  corresponding  triedrals  are  also  equal ;  and  if  two 
spherical  triangles  are  symmetrical,  their  corresponding 
triedrals  are  symmetrical. 

7*68.  Corollary. — On  the  same  sphere,  or  on  equal 
spheres,  equal  triedrals  at  the  center  have  equal  corre- 
sponding spherical  triangles  ;  and  symmetrical  triedrnls 
at  the  center  have  symmetrical  corresponding  spherical 
triangles. 

789.  Corollary — The  three  sides  and  the  three  an- 
gles of  a  spherical  triangle  are  respectively  the  measures 
of  the  three  faces  and  the  three  diedrals  of  the  triedral 
at  the  center. 

770.  Corollary Spherical   triangles    are    isosceles, 

equilateral,  rectangular,  birectangular,  and  trirectangu- 
iar,  according  to  their  triedrals. 

771.  Corollary. — The  sum  of  the  angles  of  a  spher- 
ical triangle  is  greater  than  two,  and  less  than  six  right 
angles  (591). 

772.  Corollary An   isosceles  spherical   triangle   is 


SPHERES.  259 

equal  to  its   symmetrical,  and  has  equal   angles   oppo- 
site the  equal  sides  (594). 

773.  Corollary The   radius    being    the    same,   two 

spherical  triangles  are  equal, 

1st.  When  they  have  two  sides  and  the  included  an- 
gle of  the  one  respectively  equal  to  those  parts  of  the 
other,  and  similarly  arranged; 

2d.  When  they  have  one  side  and  the  adjacent  angles 
of  the  one  respectively  equal  to  those  parts  of  the  other, 
and  similarly  arranged; 

3d.  When  the  three  sides  are  respectively  equal,  and 
similarly  arranged; 

4th.  When  the  three  angles  are  respectively  equal, 
and  similarly  arranged. 

774.  Corollary. — In  each  of  the  four  cases  just  given, 
when  the  arrangement  of  the  parts  is  reversed,  the  tri- 
angles are  symmetrical. 

POLAR    TRIANGLES. 

775.  If  at  the  vertex  of  a  triedral,  a  perpendicular 
bo  erected  to  each  face,  these  lines  form  the  edges  of  a 
supplementary  triedral  (590).  If  the  given  vertex  is  at 
the  center  of  a  sphere,  then  there  are  two  spherical  tri- 
angles corresponding  to  these  two  triedrals,  and  they 
have  all  those  relations  which  have  been  demonstrated 
concerning  supplementary  triedrals. 

Since  each  edge  of  one  triedral  is  perpendicular  to 
the  opposite  face  of  the  other,  it  follows  that  the  vertex 
of  each  angle  of  one  triangle  is  the  pole  of  the  opposite 
side  of  the  other.  Hence,  such  triangles  are  called 
polar  triangles,  though  sometimes  supplementary. 

776.  Theorem. — If  with  the  several  vertices  of  a  spher- 
ical triangle  as  poles,  arcs  of  great  circles  be  made,  then  a 


260 


ELEMENTS    OF    GEOMETRI. 


second  trianglt  is  formed  whose  vertices  are  also  poles  of 
the  first. 

777.  Theorem — Uach  angle  of  a  spherical  triangle  is 
the  supplement  of  the  opposite  side  of  its  polar  triangle. 

Let  ABC  be  the  given  triangle,  and  EF,  DF,  and  DE 
be  arcs  of  great  circles,  whose 
poles  are  respectively  A,  B, 
andC.  Then  ABC  and  DEF 
are  polar  or  supplementary- 
triangles. 

These  two  theorems  are 
corollaries  of  Article  589,  but 
they  can  be  demonstrated  by 
the  student,  with  the  aid  of 
the  above  diagram,  without  reference  to  the  triedrals. 

778.  The  student  w^ill  derive  much  assistance  from 
drawing  the  diagrams  on  a  globe.  Draw  the  polar  tri- 
angle of  each  of  the  following :  a  birectangular  triangle, 
a  trirectangular  triangle,  and  a  triangle  with  one  side 
longer  than  a  quadrant  and  the  adjacent  angles  very 
acute. 


INSCRIBED    AND    CIECUMSCR  I  BE  D. 


779.  A  sphere  is  said  to  be  inscribed  in  a  polyedron 
when  the  faces  are  tangent  to  the  curved  surface,  in  which 
case  the  polyedron  is  circumscribed  about  the  sphere.  A 
sphere  is  circumscribed  about  a  polyedron  when  the  ver- 
tices all  lie  in  the  curved  surface,  in  which  case  the  poly- 
edron is  inscribed  in  the  sphere. 

780.  Problem — Any  tetraedron  may  have  a  sphere 
inscribed  in  it;  also,  one  circumscribed  about  it. 

For  within  any  tetraedron,  there  is  a  point  equally 
distant  from  all  the  faces  (625),  which  may  ^^  'he  cen- 


SPHERICAL    AREAS. 


261 


ter  of  the  inscribed  sphere,  the  radius  beiiig  the  perpen- 
dicular distance  from  this  center  to  either  face.  There 
is  also  a  point  equally  distant  from  all  the  vertices  of 
any  tetraedron  (623),  which  may  be  the  center  of  the 
circumscribed  sphere,  the  radius  being  the  distance  from 
this  point  to  either  vertex. 

781.  Corollary. — A  spherical  surface  may  be  made  to 
pass  through  any  four  points  not  in  the  same  plane. 

EXERCISES. 

782. — 1.  In  a  spherical  triangle,  the  greater  side  is  opposite 
the  greater  angle;  and  conversely. 

2.  If  a  plane  be  tangent  to  a  sphere,  at  a  point  on  the  circum- 
ference of  a  section  made  by  a  second  plane,  then  the  intersection 
of  these  planes  is  a  tangent  to  that  circumference. 

3.  When  two  spherical  surfaces  intersect  each  other,  the  lin 
of  intersection  is  a  circumference  of  a  circle;  and  the  straight  line 
which  joins  the  centers  of  the  spheres  is  the  axis  of  that  circle. 

SPHERICAL    AREAS. 


783.  Let  AHF  be  a  right  angled  triangle  and  BFD 
a  semicircle,  the  hypotenuse  AF  be- 
ing a  secant,  and  the  vertex  F  in 
the  circumference.  From  E,  the 
point  where  AF  cuts  the  arc,  let  a 
perpendicular  EG  fall  upon  AB. 

Suppose  the  whole  of  this  figure 
to  revolve  about  AD  as  an  axis. 
The  triangle  AFH  describes  a  cone, 
the  trapezoid  EGHF  describes  the 
frustum  of  a  cone,  and  the  semicir- 
cle describes  a  sphere. 

The   points   E  and  F  describe  the  circumferences  of 


262  ELEMENTS   OF   GEOMETRY. 

the  bases  of  tke  frustum ;  and  these  circumferences  lie 
in  the  surface  of  the  sphere. 

A  frustum  of  a  cone  is  said  to  be  inscribed  in  a 
sphere,  when  the  circumferences  of  its  bases  lie  in  the 
surface  of  the  sphere. 

784.  Theorem. —  The  area  of  the  curved  surface  of  an 
inscribed  frustum  of  a  cone,  is  equal  to  the  product  of  the 
altitude  of  the  frustum  by  the  circumference  of  a  circle 
whose  radius  is  the  perpendiciilar  let  fall  from  the  center 
of  the  sphere  upon  the  slant  hirjht  of  the  frustum. 

Let  AEFD    be    the    semicircle   which   describes    the 
given  sphere,  and  EBHF  the  trape- 
zoid   Avhich    describes    the    frustum.  ^ 
Let  IC  be  the  perpendicular  let  fall              /ff 
from  the  center  of  the  sphere  upon          yiX 
the  slant  hight  EF.                                       ^/       kX 

Then  the  circumference  of  a  circle         1 
of  this  radius  would  be  tt  times  twice  \ 

CI,  or  2;rCI ;  and  it  is  to  be  proved  \ 

that  the  area  of  the  curved  surface  ^--^ 

of  the  frustum  is  equal  to  the  prod- 
uct of  BH  by  2.tCL 

The  chord  EF  is  bisected  at  the  point  I  (187).  From 
this  point,  let  a  perpendicular  IG  fall  upon  the  axis  AD. 
The  point  I  in  its  revolution  describes  the  circumference 
of  the  section  midway  between  the  two  bases  of  the 
frustum.  GI  is  the  radius  of  this  circumference,  which  is 
therefore  27rGI.  The  area  of  the  curved  surface  of  the 
frustum  is  equal  to  the  product  of  the  slant  hight  by 
this  circumference  (727);  that  is,  EF  by  2-GI. 

Now  from  E,  let  fall  the  perpendicular  EK  upon  FH. 
The  triangles  EFK  and  IGC,  having  their  sides  respect- 
ively perpendicular  to  each  other,  are  similar.  Therefore, 
EF  :  EK  :  :  CI  :  GI.     Substituting  for  the  second  term 


SPHERICAL    AREA&.  263 

EK   its  'equal  BH,  and  for  the  second   ratio  its  equi- 
multiple 2;rCI  :  2;:GI,  we  have 

EF  :  BH  :  :  2;rCI  :  2;rGI. 

Bj  multiplying  the  means  and  the  extremes, 

EFx2;ria-:BHX2;rIC. 

But  the  first  member  of  this  equation  has  been  shown 
to  be  equal  to  the  area  of  the  curved  surface  of  the 
frustum.  Therefore,  the  second  is  equal  to  the  same 
area. 

785.  Corollary. — If  the  vertex  of  the  cone  we're  at 
the  point  A,  the  cone  itself  would  be  inscribed  in  the 
sphere;  and  there  would  be  the  same  similarity  of  tri- 
angles, and  the  same  reasoning  as  above.  It  may  be 
shown  that  the  curved  surface  of  an  inscribed  cone  is 
equal  to  the  product  of  its  altitude  by  the  circumfer- 
ence of  a  circle  whose  radius  is  a  perpendicular  let  fall 
from  the  center  of  the  sphere  upon  the  slant  hight. 

•786.  Theorem — The  area  of  the  surface  of  a  sphere 
is  equal  to  the  product  of  the  diameter  hy  the  circumfer- 
ence of  a  great  circle. 

Let  ADEFGB  be  the  semicircle  by  which  the  sphere 
is  described,  having  inscribed    in   it 
half  of  a  regular  polygon  which  may  ^ 

be  supposed  to  revolve  with  it  about  //       "^^ 

the  common  diameter  AB.  /^ 

"Of. 

Then,  the  surface  described  by  the         / 
side  AD  is    equal   to  2;rCI  by  AH.         1 
The     surface    described    by    DE    is         \ 
equal  to  2;rCI  by  HK,  for  the  per-  \^— 

pendicular  let  fall  upon  DE  is  equal  ^^^^^Ub 

to    CI;    and   so    on.      If  one   of  the 
sides,  as  EF,  is  parallel  to  the  axis,  the  measure  is  the 
same,  for  the  surface  is  cylindrical.     Adding  these  sev- 


264  ELEMENTS    OF    GEOMETRY. 

eral  equations  together,  we  find  that  the  entire  surface 
described  by  the  revolution  of  the  regular  polygon  about 
its  diameter,  is  equal  to  the  product  of  the  circumfer- 
ence whose  radius  is  CI,  by  the  diameter  AB. 

This  being  true  as  to  the  surface  described  by  the 
perimeter  of  any  regular  polygon,  it  is  therefore  true 
of  the  surface  described  by  the  circumference  of  a  cir- 
cle. But  this  surface  is  that  of  a  sphere,  and  the  radius 
CI  then  becomes  the  radius  of  the  sphere.  Therefore, 
the  area  of  the  surface  of  a  sphere  is  equal  to  the 
product  of  the  diameter  by  the  circumference  of  a  great 
circle. 

787.  Corollary. — The  area  of  the  surface  of  a  sphere 
is  four  times  the  area  of  a  great  circle.  For  the  area 
of  a  circle  is  equal  to  the  product  of  its  circumference 
by  one-fourth  of  the  diameter. 

788.  Corollary — The  area  of  the  _,-_ 

surface  of  a  sphere  is  equal  to  the  inT^SIillS^^ 

area  of  the  curved  surface  of  a  cir-  |ii||iliii'P^^^^^ 

cumscribing  cylinder;  that  is,  a  cyl-  llijjr^.... jti 

inder  whose  bases  are  tangent  to  the  iiiir~~hl^^^ 

surfa.ce  of  the  sphere.  PliiSiiill 


AREAS    OF    ZONES. 

789.  A  Zone  is  a  part  of  the  surface  of  a  sphere 
included  between  two  parallel  planes.  That  portion  of 
the  sphere  itself,  so  inclosed,  is  called  a  segment.  The 
circular  sections  are  the  bases  of  the  segment,  and  the 
distance  between  the  parallel  planes  is  the  altitude  of 
the  zone  or  segment. 

One  of  the  parallel  planes  may  be  a  tangent,  in 
which  case  the  segment  has  one  base. 


SPHERICAL    AREAS. 


205 


TOO.  Theorem — The  area  of  a  zone  is  equal  to  the 
product  of  its  altitude  hy  the  circumference  of  a  great 
circle. 

This  is  a  corollary  of  the  last  demonstration  (786). 
The  area  of  the  zone  described  by  the  arc  AD,  is  equal 
to  the  product  of  AH  by  the  circumference  whose  ra- 
dius is  the  radius  of  the  sphere. 


AREAS    OF    LUXES. 

TOl.  Theorem. —  Tlie  area  of  a  lune  is  to  the  area  of 
the  whole  spherical  surface  as  the  angle  of  the  lune  is  to 
four  right  angles. 

It  has  already  been  shown  that  the  angle  of  the  lune 
is  measured  by  the  arc  of  a  great 
circle  whose  pole  is  at  the  vertex. 
Thus,  if  AB  is  the  axis  of  the  arc 
DE,  then  DE  measures  the  angle 
DAE,  which  is  equal  to  the  angle 
DCE.  But  evidently  the  lune  varies 
exactly  with  the  angle  DCE  or  DAE. 
This  may  be  rigorously  demonstrated 
in  the  same  manner  as  the  principle 
that  angles  at  the  center  have  the  same  ratio  as  their 
intercepted  arcs. 

Therefore,  the  area  of  the  lune  has  the  same  ratio  to 
the  whole  surface  as  its  angle  has  to  the  whole  of  four 
right  angles. 


TRIRECTANGULAR    TRIANGLE. 

T02.  If  the  planes  of  two  great  circles  are  perpen- 
dicular to  each  other,  they  divide  the  surface  into  four 
equal  lunes.     If  a  third  circle  be  perpendicular  to  these 
Geom.— 23 


266 


ELEMENTS    OF   GEOMETRY. 


two,  each  of  the  four  lunes  is  divided  into  two  equal 
triangles,  which  have  their  angles  all  right  angles  and 
their  sides  all  quadrants.  Hence,  this  is  sometimes 
called  the  quadrantal  triangle. 

This  triangle  is  the  eighth  part  of 
the  whole  surface,  as  just  shown.  Its 
area,  therefore,  is  one-half  that  of  a 
great  circle  (787).  Since  the  area  of 
the  circle  is  ;r  times  the  square  of 
the  radius,  the  area  of  a  trirectangu- 
lar  triangle  may  be  expressed  by  J;rR^. 

The  area  of  the  trirectangular  triangle  is  frequently 
assumed  as  the  unit  of  spherical  areas. 


AREAS    OF    SPHERICAL    TRIANGLES. 

793.  Theorem. — Two  symmetncal  spherical  trianales 
are  equivalent. 

Let  the  angle  A  be  equal  to  B,  E  to  C,  and  I  to  B. 
Then  it  is  known  that 
the  other  parts  of  the  ^ 

triangle  are  respect- 
ively equal,  but  not 
superposable ;  and  it 
is  to  be  proved  that 
the  triangles  are  equiv- 
alent. 

Let  a  plane  pass  through  the  three  points  A,  E,  and 
I ;  also,  one  through  B,  C,  and  D.  The  sections  thus  made 
are  small  circles,  which  are  equal;  since  the  distances 
between  the  given  points  are  equal  chords,  and  circles 
described  about  equal  triangles  must  be  equal.  Let  0 
be  that  pole  of  the  first  circle  which  is  on  the  same 
gide    of   \\io,    S2)hcre    as   the  triangle,  and    F    the   corre- 


SPHERICAL    AREAS.  2G7 

spending  pole  of  the  second  small  circle.  Let  0  be 
joined  bj  arcs  of  great  circles  OA,  OE,  and  01,  to  the 
several  vertices  of  t\e  first  triangle  ;  and,  in  the  same 
way,  join  FB,  FC,  and  FD. 

Now,  the  triangles  AOI  and  BFD  are  isosceles,  and 
mutually  equilateral ;  for  AO,  10,  BF,  and  DF  are  equal 
arcs  (753).  Hence,  these  triangles  are  equal  (772). 
For  a  similar  reason,  the  triangles  lOE  and  CFD  are 
equal ;  also,  the  triangles  AOE  and  BFC.  Therefore, 
the  triangles  AEI  and  BCD,  being  composed  of  ecual 
parts,  are  equivalent. 

The  pole  of  the  small  circle  may  be  outside  of  the 
given  triangle,  in  which  case  the  demonstration  would 
be  by  subtracting  one  of  the  isosceles  triangles  from  the 
sum  of  the  other  two. 

•704.  It  has  been  show^n  that  the  sum  of  the  angles 
of  a  spherical  triangle  is  greater  than  the  sum  of  the 
angles  of  a  plane  triangle  (771).  Since  any  spherical 
polygon  can  be  divided  into  triangles  in  the  same  man- 
ner as  a  plane  polygon,  it  follows  that  the  sum  of  the 
angles  of  any  spherical  polygon  is  greater  than  the  sum 
of  the  angles  of  a  plane  polygon  of  the  same  number 
of  sides. 

The  difference  betw^een  the  sum  of  the  angles  of  a 
spherical  triangle,  or  other  polygon,  and  the  sum  of  the 
angles  of  a  plane  polygon  of  the  same  number  of  sides, 
is  called  the  spherical  excess. 

795.  Theorem. —  The  area  of  a  spherical  triangle  is 
equal  to  the  area  of  a  trirect angular  triangle,  multiplied 
hy  the  ratio  of  the  spherical  excess  of  the  given  triangle  to 
one  right  angle. 

That  is,  the  area  of  the  given  triangle  is  to  that  of 
the  trirectangular  triangle,  as  the  spherical  excess  of  the 
given  triangle  is  to  one  right  angle. 


268 


ELEMEJ^TS   OF   GEOMETRY. 


Let  AEI  be  any  spherical  triangle,  and  let  DHBCGF 
be  any  great  circle,  on  one  side  of  which  is  the  given 
triangle.  Then,  comsidering  this  circle  as  the  plane  of 
reference  of  the  figure,  produce  the  sides  of  the  trian- 
gle AEI  around  the  sphere. 

Now,  let  the  several  angles  of  the  given  triangle  be 
represented  by  a,  e,  and  i;  that  is,  taking  a  right  an- 
gle for  the  unit,  the  angle  EAI  is  equal  to  a  right 
angles,  etc.  Then,  the  area 
of  the  lune  AEBOCI  is  to 
the  whole  surface  as  a  is  to 
4  (791).  But  if  the  tri- 
rectangular  triangle,  which 
is  one-eighth  of  the  spher- 
ical surface,  be  taken  as 
the  unit  of  area,  then  the 
area  of  this  lune  is  2a. 
But  the  triangle  BOC,  which 
is  a  part  of  this  lune,  is  equivalent  to  its  opposite  and 
symmetrical  triangle  DAF.  Substituting  this  latter, 
the  area  of  the  two  triangles  ABC  and  DAF  is  2a  times 
the  unit  of  area. 

In  the  same  way,  show  that  the  area  of  the  tw^o  tri- 
angles IDH  and  IGC  is  2z,  and  that  the  area  of  the 
two  triangles  EFG  and  EHB  is  2e  times  the  unit  of 
area.      These  equations  may  be  wTitten  thus : 

area  (ABO  +  ADF)  =  2a  times  the  trirectangular  tri- 
angle ; 

area  ( IDH  +  IGC )  =  2i  times  the  trirectangular  tri- 
angle ; 

area  (EFG  +  EHB)  =  2e  times  the  trirectangular  tri- 
angle. 

In  adding  these  equations  together,  take  notice  that 
the  triangles  mentioned  include  the  given  triangle  AEI 


SPHERICAL    AREAS.  269 

three  times,  and  all  the  rest  of  the  surface  of  the  hem- 
isphere above  the  plane  of  reference  once ;  also,  that 
the  area  of  this  hemispherical  surface  is  four  times  that 
of  the  trirectangular  triangle.  Then,  by  addition  of  tlio 
equations  : 
area  4  trirect.  tri.  +  2  area  AEI  =  2  («-f  e-{-i)  trir.  Iri. 

Transposing  the  first  term,  and  dividing  by  2. 
area  AEl  =  (a -]- e -\r  i — 2)  trir.  tri. 

But  (a-i-e-i-i — 2)  is  the  spherical  excess  of  the 
given  triangle,  taking  a  right  angle  as  a  unit ;  that  is,  it 
is  the  ratio  of  the  spherical  excess  of  the  given  trian- 
gle to  one  right  angle. 

796.  Corollary. — If  the  square  of  the  radius  be  taken 
as  the  unit  of  area,  then  the  area  of  any  spherical  tri- 
angle may  be  expressed  (792), 

J(a  +  e  +  ^  — 2};rR2. 

AREAS    OF    SPHERICAL    POLYGONS. 

TOy.  Theorem — The  area  of  any  spherical  polygon  is 
equal  to  the  area  of  (he  trirectangular  triangle  multiplied 
hy  the  ratio  of  the  spherical  excess  of  the  polygon  to  one 
right  angle. 

For  the  spherical  excess  of  the 
polygon  is  evidently  the  sum  of 
the  spherical  excess  of  the  trian- 
gles which  compose  it;  and  its 
area  is  the  sum  of  their  areas. 

EXERCISES. 

198. — 1.  What  is  the  area  of  the  earth's  surface,  supposing  it 
to  be  in  the  shape  of  a  sphere,  with  a  diameter  of  7912  miles? 


270  ELEMENTS   OF    GEOMETllY. 

2.  Upon  the  same  hypothesis,  what  portion  of  the  whole  sur- 
face is  between  the  equator  and  the  parallel  of  30°  north  latitude? 

3.  Upon  the  same  hypothesis,  what  portion  of  the  whole  sur- 
face is  between  two  meridians  which  are  ten  degrees  apart? 

4.  What  is  the  area  of  a  triangle  described  on   a  globe  of  13 
inches  diameter,  the  angles  being  100°,  45°,  and  53°? 


VOLUME    OF    THE    SPHEEE. 

•700.  Theorem. —  The  vohijne  of  any  poly edr on  in  which 
a  sphere  can  be  inscribed^  is  equal  to  one-third  of  the  prod- 
uct of  the  entire  surface  of  the  polyedron  by  the  radius  of 
the  inscribed  sphere. 

For,  if  a  plane  pass  through  each  edge  of  the  poly- 
edron, and  extend  to  the  center  of  the  sphere,  these 
planes  will  divide  the  polyedron  into  as  many  pyramids 
as  the  figure  has  faces.  The  faces  of  the  polyedron  are 
the  bases  of  the  pyramids. 

The  altitude  of  each  is  the  radius  of  the  sphere,  for 
the  radius  which  extends  to  the  point  of  tangency  is 
perpendicular  to  the  tangent  plane  (742).  But  the  vol- 
ume of  each  pyramid  is  one-third  of  its  base  by  its 
altitude.  Therefore,  the  volume  of  the  Avhole  polyedron 
IS  one-third  the  sum  of  the  bases  by  the  common  alti- 
tude, or  radius. 

800,  Theorem. —  The  volume  of  a  sphere  is  equal  to 
one-third  of  the  product  of  the  surface  by  the  radius. 

For,  the  surface  of  a  sphere  may  be  approached  as 
nearly  as  Ave  choose,  by  increasing  the  number  of  faeej 
of  the  circumscribing  polyedron,  until  it  is  evident  that 
the  sphere  is  the  limit  of  the  polyedrons  in  which  it  is 
inscribed.  Then,  this  theorem  becomes  merely  a  corol- 
lary of  the  preceding. 

801.  Corollary. — The  volume  of  a  spherical  pyramid, 


SPHERICAL    VOLUMES. 


271 


or  of  a  spherical  wedge,  is  equal   ^;o  one-third   of  the 
product  of  its  spherical  surface  by  the  radius. 

802.  A  spherical  Sector  is  that  portion  of  a  sphere 
which  is  described  by  the  rev- 
olution of  a  circular  sector 
about  a  diameter  of  the  circle. 
It  may  have  two  or  three  curved 
surfaces. 

Thus,  if  AB  is  the  axis,  and 
the  generating  sector  is  AEC, 
the  sector  has  one  spherical 
and  one  conical  surface ;  but  if, 
with  the  same  axis,  the  gener- 
ating sector  is  FCG,  then  the  sector  has  one  spherical 
and  two  conical  surfaces. 

803.  Corollary. — The  volume  of  a  spherical  sector  is 
equal  to  one-third  of  the  product  of  its  spherical  surface 
by  the  radius. 

804.  The  volume  of  a  spherical  segment  of  one  base 
is  found  by  subtracting  the 
volume  of  a  cone  from  that 
of  a  sector.  For  the  sector 
ABCD  is  composed  of  the 
segment  ABC  and  the  cone 
ACD. 

The  volume  of  a  spherical  segment  of  two  bases  is 
the  difference  of  the  volumes  of  two  segments  each  of 
one  base.  Thus  the  segment  AEFC  is  equal  to  the 
segment  ABC  less  EBF. 

805.  All  spheres  are  similar,  since  they  are  gener- 
ated by  circles  which  are  similar  figures.  Hence,  we 
might  at  once  infer  that  their  surfaces,  as  well  as  their 
volumes,  have  the  same  ratios  as  in  other  similar  solids. 
These  principles  may  be  demonstrated  as  follows : 


272  ELEMENTS  OF  GEOMETRY. 

800.  Theorem.  —  The  areas  of  the  surfaces  of  two 
spheres  are  to  each  other  as  the  squares  of  their  diameters; 
and  their  volumes  are  as  the  cubes  of  their  diameters,  or 
other  homologous  lines. 

For  the  superficial  area  of  any  sphere  is  equal  to  t: 
times  the  diameter  multiplied  by  the  diameter  (786); 
that  is  ;rD'^.  But  ;r  is  a  certain  or  constant  factor. 
Therefore,  the  areas  vary  as  the  squares  of  the  diame- 
ters. 

The  volume  is  equal  to  the  product  of  the  surface  by 
one-sixth  of  the  diameter  (800) ;  that  is,  TtW  by  JD, 
or  ^TiW.  But  Jtt  is  a  constant  numeral.  Therefore, 
the  volumes  vary  as  the  cubes  of  the  diameters. 

USEFUL    FORMULAS. 

SOT.  Represent  the  radius  of  a  circle  or  a  sphere, 
or  that  of  the  base  of  a  cone  or  cylinder,  by  R ;  repre- 
sent the  diameter  by  D,  the  altitude  by  A,  and  tha  slant 
hight  by  H. 

Circumference  of  a  circle      .  =  ttD  =  27rR, 

Area  of  a  circle  =  JttD^  =  ;rR'^, 

Curved  surface  of  a  cone  =  J;rDH  =  ttRH, 

Entire  surface  of  a  cone  =  ;rR(H-}-R), 

Volume  of  a  cone  =  j^^ttD^A  =  J;rR^A, 

Curved  surface  of  a  cylinder  =  ;rDA  =  2;rRA, 

Entire  surface  of  a  cylinder  =  27rR(A  +  Rj. 

Volume  of  a  cylinder  =  J;rD-A  =  ;rR^A, 

Surface  of  a  sphere  =;rD-=4;rR^, 

Volume  of  a  sphere  =  iyTD'^=|;rR^, 
;:  =  3.1415926535. 


EXERCISES    FOR  -GENERAL    REVIEW.  273 


EXERCISES. 

SOS, — 1.  What  is  the  locus  of  those  points  in  space  which  are 
at  the  same  distance  from  a  given  point? 

2.  What  is  the  locus  of  those  points  in  space  which  are  at  the 
samo  distance  from  a  given  straight  line? 

3.  Wliat  is  the  locus  of  those  points  in  space,  such  that  the 
distance  of  each  from  a  gi\^en  straight  line,  has  a  constant  ratio 
to  its  distance  from  a  given  point  of  that  line? 


EXERCISES    FOR    GENERAL    REVIEW. 

809. — I.  Take  some  principle  of  general  application,  and  state 
all  its  consequences  which  are  found  in  the  chapter  under  review; 
arranging  as  the  first  class  those  which  are  immediately  deduced 
from  the  given  principle;  then,  those  which  are  derived  from 
these,  and  so  on, 

2.  Reversirng  the  above  operation,  take  some  theorem  in  the 
latter  part  of  a  chapter,  state  all  the  principles  upon  which  its 
proof  immediately  depends;  then,  all  upon  which  these  depend; 
and  so  on,  back  to  the  elements  of  the  science. 

3.  Given  the  proportion,  a  :  h  :'.  c  :  d, 
to  show  that  c  —  a  :  d — h  :  :  a  :  b ; 

also,  that  a-\-c  :  a  —  c  :  :  b-^d  :  b — d. 

4.  Form  other  proportions  by  combining  the  same  terms. 

5.  What  is  the  greatest  number  of  points  in  which  seven 
straight  lines  can  cut  each  other,  three  of  them  being  parallel; 
and  what  is  the  least  number,  all  the  lines  being  in  one  plane? 

6.  If  two  opposite  sides  of  a  parallelogram  be  bisected,  straight 
lines  from  the  points  of  bisection  to  the  opposite  vertices  will  tri- 
sect the  diagonal. 

7.  In  any  triangle  ABC,  if  BE  and  CF  be  perpendiculars  to 
any  line  through  A,  and  if  D  be  the  middle  of  BC,  then  DE  is 
equal  to  DF. 

8.  If,  from  the  vertex  of  the  right  angle  of  a  triangle,  there 
extend  two  lines,  one   bisecting   the   base,  and    the  other  perpen- 


274  ELEMENTS  OF  GEOMETRY. 

dicular  to    it,  the  angle  of  these  two  lines  is  equal  to  the  differ- 
ence of  the  two  acute  angles  of  the  triangle. 

9.  In  the  base  of  a  triangle,  find  the  point  from  which  lines 
extending  to  the  sides,  and  parallel  to  them,  will  be  equal. 

10.  To  construct  a  square,  having  a  given  diagonal. 

11.  Two  triangles  having  an  angle  in  the  one  equal  to  an 
angle  in  the  other,  have  their  areas  in  the  ratio  of  the  products 
of  the  sides  including  the  equal  angles. 

12.  If,  of  the  four  triangles  into  which  the  diagonals  divide  a 
quadrilateral,  two  opposite  ones  are  equivalent,  the  quadrilateral 
has  two  opposite  sides  parallel. 

13.  Two  quadrilaterals  are  equivalent  when  their  diagonals  are 
respectively  equal,  and  form  equal  angles. 

14.  Lines  joining  the  middle  points  of  the  opposite  sides  of  any 
quadrilateral,  bisect  each  other. 

15.  Is  there  a  point  in  every  triangle,  such  that  any  straight 
line  through  it  divides  the  triangle  into  equivalent  parts? 

16.  To  construct  a  parallelogram  having  the  diagonals  and 
one  side  given. 

17.  The  diagonal  and  side  of  a  square  have  no  common  meas- 
ure, nor  common  multiple.  Demonstrate  this,  without  using  the 
algebraic  theory  of  radical  numbers. 

18.  To  construct  a  triangle  when  the  three  altitudes  are  given. 

19.  To  construct  a  triangle,  when  the  altitude,  the  line  bisect- 
ing the  vertical  angle,  and  the  line  from  the  vertex  to  the  middle 
of  the  base,  are  given. 

20.  If  from  the  three  vertices  of  any  triangle,  straight  lines 
be  extended  to  the  points  where  the  inscribed  circle  touches  the 
Bides,  these  lines  cut  each  other  m  one  point. 

21.  What  is  the  area  of  the  sector  whose  arc  is  50°,  and  whose 
radius  is  10  inches? 

22.  To  construct  a  square  equivalent  to  the  sum,  or  to  the  diii 
ference  of  two  given  squares. 

23.  To  divide  a  given  straight  line  in  the  ratio  of  the  areas  of 
two  given  squares. 

24.  If  all  tlie  sides  of  a  polygon  except  one  be  given,  its  area 
will  be  greatest  when  the  excepted  side  is  made  the  diameter  of  a 
circle  which  circumscribes  the  polygon. 


EXERCISES    FOR    GENERAL   REVIEW.  275 

25.  Find  the  locus  of  those  points  in  a  plane,  such  that  the 
sum  of  the  squares  of  the  distances  of  each  from  two  given  points, 
shall  be  equivalent  to  the  square  of  a  given  line. 

26.  Find  the  locus  of  those  points  in  a  plane,  such  that  the 
difference  of  the  squares  of  the  distances  of  each  from  two  given 
points,  shall  be  equivalent  to  the  square  of  a  given  line. 

27.  If  the  triangle  DEF  be  inscribed  in  the  triangle  ABC,  the 
circumferences  of  the  circles  circumscribed  about  the  three  trian- 
gles AEF,  BFD,  CDE,  will  pass  through  the  same  point. 

28.  The  three  points  of  meeting  mentioned  in  Exercises  28,  29, 
and  30,  Article  337,  are  in  the  same  straight  Jine. 

29.  If,  on  the  sides  of  a  given  plane  triangle,  equilateral  tri- 
angles be  constructed,  the  triangle  formed  by  joining  the  centers 
of  these  three  triangles  is  also  equilateral;  and  the  lines  joining 
their  vertices  to  the  opposite  vertices  of  the  given  triangle  are 
equal,  and  intersect  in  one  point. 

30.  The  feet  of  the  three  altitudes  of  a  triangle  and  the  cen- 
ters of  the  three  sides,  all  lie  in  one  circumference.  The  circle 
thus  described  is  known  as  "  The  Six  Points  Circle." 

31.  Four  circles  being  described,  eacli  of  which  shall  touch  the 
three  sides  of  a  triangle,  or  those  sides  produced ;  if  six  lines  be 
made,  joining  the  centers  of  those  circles,  two  and  two,  then  the 
middle  points  of  these  six  lines  are  in  the  circumference  of  the 
circle  circumscribing  the  given  triangle. 

32.  If  two  Imes,  one  being  in  each  of  two  intersecting  planes, 
are  parallel  to  each  other,  then  both  are  parallel  to  the  intersec- 
tion ff  the  planes. 

33.  If  a  line  is  perpendicular  to  one  of  two  perpendicular 
planes,  it  is  parallel  to  the  other;  and,  conversely,  if  a  line  is  par- 
allel to  one  and  perpendicular  to  another  of  two  planes,  then  the 
planes  aro  perpendicular  to  each  other. 

34.  How  mny  a  pyramid  be  cut  by  a  plane  parallel  to  the  base, 
■.0  as  to  make  the  area  or  the  volume  of  the  part  cut  off  have  a 
£;iven  ratio  to  the  area  or  the  volume  of  the  whole  pyramid? 

35.  Any  regular  polyedron  may  have  a  sphere  inscribed  in  it; 
also,  one  circumscribed  about  it 

36  In  any  polyedron,  the  sum  of  the  number  of  vertices  and 
the  number  of  faces  exceeds  by  two  the  number  of  edges. 


276  ELEMENTJS   OF   GEOMETRY. 

37.  How  many  spheres  can  be  made  tangent  to  three  given 
planes? 

38.  Apply  to  spheres  the  principle  of  Article  331;  also,  of 
Article  191,  substitutmg  circles  for  chords. 

39.  Discuss  the  possible  relative  positions  of  two  spheres. 

40.  What  is  the  locus  of  those  points  in  space,  such  that  the 
sum  of  the  .squares  of  the  distances  of  each  from  two  given 
points,  is  equivalent  to  a  given  square? 

41.  What  is  the  locus  of  those  points  in  space,  such  that  the 
difference  of  the  squares  of  the  distances  of  each  from  two  given 
points,  is  equivalent  to  a  given  square  ? 

42.  A  frustum  of  a  pyramid  is  equivalent  to  tlie  s.um  of  three 
pyramids  all  having  the  same  altitude  as  the  frustum,  and  having 
lor  their  bases  the  lower  base  of  the  frustum,  the  upper  base,  and 
a  mean  proportional  between  them. 

43.  The  surface  of  a  sphere  can  be  completely  covered  with  tlie 
surfaces  either  of  4,  or  of  8,  or  of  20  equilateral  spherical  tri- 
angles. 

44.  The  volume  of  a  cone  is  equal  to  the  product  of  its  whole 
surface  by  one-third  the  radius  of  the  inscribed  sphere. 

45.  If,  about  a  sphere,  a  cylinder  be  circumscribed,  also  a  cone 
whose  slant  hight  is  equal  to  the  diameter  of  its  base,  tlien  the 
area  and  volume  of  the  sphere  are  two-thirds  of  the  area  and 
volume  of  the  cylinder;  and  the  area  and  volume  of  the  cylinder 
are  two-thirds  of  the  area  and  volume  of  the  cone. 


TRIGONOMETRY. 

CHAPTER  XII. 
PLANE    TRIGONOMETRY. 

811.  Trigonometry  is  the  science  in  which  the  rehi- 
tions  subsisting  between  the  angles,  sides,  and  area  of 
any  triangle  are  investigated.  The  science  was  origi- 
nally called  Plane  Trigonometry  or  Spherical  Trigonom- 
etry, according  as  the  triangle  was  plane  or  spherical. 

Plane  Trigonometry  has  now  a  wider  meaning,  com- 
prising algebraic  investigations  concerning  angles  and 
their  functions,  and  the  methods  of  calculating  these 
functions. 

MEASURE    OF   ANGLES. 

812.  In  Elementary  Geometry,  the  unit  for  the  meas- 
ure of  angles  is  usually  the  right  angle.  The  frequent 
fractions  which  the  use  of  this  unit  gives  rise  to,  render 
it  inconvenient  for  calculation.  It  has  been  divided  into 
degrees,  minutes,  and  seconds  (208). 

This  sexagesimal  division  of  angles  has  been  in  use 
since  the  second  century.  Efforts  have  been  made  to 
substitute  for  it  the  centesimal  division,  making  the  right 
angle  contain  one  hundred  grades^  each  grade  one  hun- 
dred minutes^  and  so  on ;  but  this  plan  has  never  been 

generallv  in  use. 

(277) 


278 


PLANE  trigonoml:tiiy. 


81S.  There  is  another  unit  which  has  been  called  the 
circular  measure  of  an  angle.  It  is  used  in  trigonometri- 
cal investigation,  and  is  also  called  the  analytical  unit* 
It  is  that  angle  at  tiie  center  of  a  circle 
whose  intercepted  arc  has  the  same  lin- 
ear extent  as  the  radius.  Thus,  if  the 
arc  AB  has  the  same  linear  extent  as 
the  radius  AC,  then  the  angle  C  is  the 
unit  of  circular  measure.  Hence,  this 
unit  of  measure  is  equal  to 


180° 


57°.  29578— =57°  17'  44'^  8+, 


Also,    1°=tt7q=.017453  times  the  circular  measure. 


814.  Various  instruments  are  used  for  the  measure 
of  angles.  A  protractor  is  used  to  measure  the  angle  of 
two  lines  in  a  drawing.  It  is  usually  shaped  like  a  semi- 
circumference  with  its  diameter,  the  arc  being  marked 
with  the  degrees  from  0  to  180. 

Let  it  be  required  to  measure  the  angle  ABC.  Place 
the  center  of  the  straight 
edge,  which  is  marked  by  a 
notch  on  the  instrument,  at 
the  vertex  B;  let  the  edge 
lie    along   one   side   of   the 

angle,  as  BC ;  then  read  the  degree  marked  where  the 
other  side  BA  passes  the  arc  of  the  instrument.  This 
gives  the  size  of  the  angle. 

The  same  instrument  is  used  for  drawing  angles  of  a 
known  size.  One  side  of  the  angle  being  drawn,  place 
the  center  of  the  protractor  at  the  point  which  is  to  be 
the  vertex;  then  the  required  number  of  degrees,  on  the 


FUNCTIONS  OF  ANGLES.  279 

edge  of  the  arc,  ^vill  indicate  a  point  on  the  other  side  of 
the  angle.  Connect  this  point  with  the  vertex  to  com- 
plete the  angle. 

The  student  should  be  provided  with  a  protractor,  a 
six-inch  scale,  and  a  pair  of  dividers.  Large  protractors, 
made  of  wood,  pasteboard,  or  tin-plate,  are  useful  for 
blackboard  work. 

EXERCISES. 

815. 1.  Find  the  circular  measure  of  an  angle  of  3°  4'  5^\ 

2.  Draw  a  triangle  having  one  side  two  inches,  another  three 
inches,  and  the  included  angle  100°.  Find  the  other  angles  and 
side  by  measurement. 

3.  Draw  a  triangle  with  the  sides  three,  four,  and  five  inches  in 
length.     Find  the  angles  by  measurement. 

These  exercises  may  be  extended  and  varied,  referring  to  Articles 
295  to  301  inclusive. 


FUNCTIONS    OF   ANGLES. 

816.  When  two  quantities  are  so  related  that  any 
variation  in  one  causes  a  variation  in  the  other,  each  is  a 
function  of  the  other.  Thus,  »Jx  is  a  function  of  x;  the 
area  of  a  circle  is  a  function  of  its  radius  (500). 

A  quantity  may  be  a  function  of  several  others.  Thus, 
x^  y^  is  a  function  of  x  and  y ;  the  area  of  a  triangle  is  a 
function  of  its  base  and  altitude  (386).  The  angles  of  a 
triangle  are  functions  of  the  ratios  of  the  sides  (316); 
and  tlie  ratios  of  the  sides  of  a  triangle  are  functions  of 
tiie  angles  (309). 

For  example,  if  the  lengths  of  the  sides  be  as  the  num- 
bers 3,  4,  and  5,  then  the  angle  opposite  the  longest  side 
is  a  right  angle  (413);  and  each  of  the  acute  angles  is 
also  a  function  of  the  numbers  3,  4,  and  5. 


200 


PLANE  TRIGONOMETRY. 


For  another  example;  if  the  tri- 
angle ACD  has  its  angles  30°,  60°, 
and  90°,  then  it  may  be  shown  that 

AC     ,    CD_  AC         .. 

AD=^'  AD-^^''''^OD=^^^-     A  D  B 

Let  the  student  now  solve  the  1st  Exercise  of  Art.  472. 

817.  Theorem — If  from  any  point  in  one  side  of  an 
ayigle,  a  perpendicular  fall  upon  the  other  side,  a  right 
angled  triangle  is  formed,  and  the  ratios  of  the  sides  of 
this  triangle  are  functions  of  the  given  angle. 

For,  if  any  number  of  tri- 
angles were  thus  formed  with 
a  given  angle,  all  of  these  tri- 
angles would  be  similar  (306), 
and  their  sides  would  have  the 
same  ratios  (309). 

When  the  given  angle  is 
greater  than  a  right  angle, 
one  side  may  be  produced  to 
meet  the  perpendicular. 


818.  If  from  any  point  on  one  side  of  a  given  angle  a 
perpendicular  fall  on  the  other  side  as  a  base,  then 

The  Sine  of  the  given  angle  is  the  ratio  of  the  perpen- 
dicular to  the  hypotenuse  of  the  right  angled  triangle  thus 
formed. 

The  Tangent  of  the  angle  is  the  ratio  of  the  perpen- 
dicular to  the  base. 

The  Secant  of  the  angle  is  the  ratio  of  the  hypotenuse 
to  the  base. 

The  Cosine  of  the  angle  is  the  ratio  of  the  base  to  the 
hypotenuse. 


FUNCTIONS  OF  ANGLES  281 

The  Cotangent  of  the  angle  is  the  ratio  of  the  base  to 
the  perpendicular. 

The  Cosecant  of  the  angle  is  the  ratio  of  the  hypote- 
nuse to  the  perpendicular. 

The  abbreviations  sin.,  tan.,  sec,  cos.,  cot.,  and  cosec. 
are  used  respectively  for  these  six  functions.  Thus,  the 
sine  of  the  angle  A  is  written  sin.  A. 

These  six  are  all  the  ratios  that  can  be  formed  by  the 
simple  combination  of  the  sides  of  the  triangle.  They 
are  called,  therefore,  the  simple  functions  of  an  angle. 
Other  functions  have  been  formed  by  composition  and  by 
division.  Of  these,  the  following  is  used  at  the  present 
day: 

The  Versed  sine  is  the  ratio  of  the  excess  of  the 
hypotenuse  over  the  base,  to  the  hypotenuse.     Hence, 

vers.  sin.  A=l — cos.  A. 

819.  A  table  of  sines  of  every  degree  from  0  to  90° 
may  be  made  by  drawing  and  measurement.  Draw  a 
right  angled  triangle,  with  an  angle  at  the  base  equal  to 
the  angle  whose  sine  w^e  wish  to  find.  It  will  simplify  the 
work  to  make  the  hypotenuse  the  length  of  a  certain  unit. 
Divide  the  length  of  the  perpendicular  by  that  of  the  hy- 
potenuse. The  quotient  is  the  sine.  By  careful  drawing 
and  measurement,  a  table  of  sines  may  be  made  that  shall 
be  true  to  two  places  of  decimals. 

A  table  of  tangents  may  be  formed  in  a  similar  manner, 
making  the  base  the  length  of  a  certain  unit. 

By  calculation,  the  functions  may  all  be  found  to  any 
required  degree  of  accuracy. 

820.  The  etymology  of  sine,  tangent,  and  secant  ap- 
pears from  the  method  which  was  formerly  used  to  define 
these  terms,  which  was  as  follows : 

Trij:.-24. 


282 


TLANE  TRIGONOMETRY. 


If  with  any  radius  an  arc  be  de- 
scribed about  the  vertex  C  as  a  cen- 
ter, and  if  from  B,  one  extremity  of 
the  intercepted  arc,  a  perpendicular 
BD  fall  upon  the  side  CA,  then  BD 
is  called  the  sine  of  the  arc  BA,  or 
of  the  angle  C.  If  a  perpendicular 
to  AC  be  produced  to  meet  CB  at  E, 
then  AE  is  the  tangent  and  CE  the 
secant  of  the  arc  AB,  or  of  the  an- 
gleC. 

The  student  readily  perceives  that  if  the  radius  is  taken 
as  the  unit  of  length,  then  the  lengths  of  BD,  AE,  and 
CE  are  respectively  the  sine,  tangent,  and  secant  of  the 
angle  C.  The  names  tangent  and  secant  are  taken  from 
the  geometrical  tangent  and  secant.  Arc,  chord,  and 
sine  are  derived  from  the  fancied  resemblance  of  the 
figure  to  the  bow  of  the  archer.  The  curve  BAF  is  the 
bow  or  arc^  the  chord  BE  joins  its  ends,  and  BD  touches 
the  breast  or  sinus^  of  the  archer.  So  also  DA  has  been 
called  the  sagitta  or  arrow.  When  used  now,  it  is  called 
the  versed  sine. 

The  oldest  work  on  Trigonometry  now  extant  is  the 
Almagest  of  Ptolemy,-  written  in  the  second  century. 
He  divides  the  radius  into  sixty  parts,  also  the  arc  whose 
chord  is  equal  to  the  radius  into  the  same  number  of 
parts.  This  mode  of  measuring  arcs,  and  consequently 
angles,  remains  in  use,  but  the  sexagesimal  division  of 
lines  was  long  since  abandoned.  The  use  of  sines  was 
introduced  by  the  Arabian  mathematicians  about  the 
eighth  or  ninth  century.  Napier,  a  Scotch  Baron,  who 
lived  in  the  early  part  of  the  seventeenth  century,  has 
done  more  for  the  science  of  Trigonometry  than  any 
other  one  man. 


FUNCTIONS  OF  ANGLES. 


283 


EXERCISES. 

821.— 1.  Demonstrate  tan.  45°=1;  also,  sin.  60°=^i/3, 
2.  Construct*  an  angle  whose  sine  is  f ;  one  whose  tangent  is  4; 
one  whose  secant  is  |. 


SIGNS  OF  ANGLES  AND  OF  THEIR  FUNCTIONS. 

822.  An  angle  may  be  conceived  to  be  generated  by 
the  revolution  of  a  line  about  a  point.  Thus,  the  line 
AB  beginning  at 
AX,  may  take  the 
positions  AB,AB', 
AB'',  AB''',  AX, 
and  so  on  indefi- 
nitely, repeating  at 
each  revolution  all 
the  positions  of  the 
first. 

In  Trigonometry,  the  amount  of  this  revolution  is  con- 
sidered as  an  angle,  so  that  an  angle  may  be  greater  than 
the  sum  of  two  or  of  four  right  angles.  In  the  strict 
geometrical  definition,  an  angle  being  the  difference  of 
two  directions,  can  not  be  greater  than  two  right  angles. 

Quantities  conceived  to  exist  in  a  certain  direction  or 
mode  are  called  positive,  and  are  designated  by  the  sign 
-|-;  while  the  quantities  in  the  opposite  direction  are  called 
negative,  and  are  designated  by  the  sign  — . 

In  the  present  investigation,  the  angle  is  supposed  to 
be  generated  by  the  motion  of  the  line  AB  up  from  AX. 
Angles  so  formed  are  positive,  and  when  estimated  in  the 
opposite  direction  they  are  negative.  Thus,  if  BAG  is 
an  acute  angle,  it  is  positive.  If  it  is  negative,  it  is 
greater  than  the  sum  of  three  right  angles.     The  com- 


284  PLANE  TRIGONOMETRY. 

plement  of  an  angle  greater  than  a  right  angle  must  be 
negative,  and  the  same  is  true  of  the  supplement  of  an 
angle  greater  than  two  right  angles. 

The  directions  to  the  right  of  YY'  and  those  upwards 
from  XX'  are  positive.  Then  the  directions  to  the  left 
from  YY'  and  those  downward  from  XX'  are  negative. 
Thus,  AC,  CB,  and  C'B'  are  positive,  while  AC,  C"B'', 
and  C'B'''  are  negative. 

823.  Theorem — The  functions  of  any  acute  angle  are 

'positive. 

For  when  the  revolving  line  is  in  the  first  quarter  of 
its  revolution,  that  is,  between  AX  and  AY,  all  the  sides 
of  the  triangle  ABC  are  positive. 

The  same  is  true  of  the  functions  of  any  angle  which 
is  equal  to  4n  right  angles  -|-  an  acute  angle,  n  being 
any  entire  number  positive  or  negative. 

824.  Theorem — The  tangent,  secant,  cosine,  and  co- 
tangent of  obtuse  angles  are  negative,  while  the  sine  and 
cosecant  of  obtuse  angles  are  positive. 

For,  when  the  revolving  line  is  in  the  second  quarter 
of  its  revolution,  that  is,  between  AY  and  AX',  the  side 
AC  of  the  triangle  AB'C  is  the  only  negative  term. 
Hence,  the  functions  of  which  it  forms  one  term  are  neg- 
ative. 

The  same  is  true  of  the  respective  functions  of  any 
angle  which  is  equal  to  an  obtuse  angle  ±  4n  right 
angles. 

82«5.  In  this  manner,  the  signs  of  the  functions  may 
be  found,  and  arranged  according  to  the  quarter  of  the 
revolving  line  AB.  The  following  table  exhibits  the 
signs  of  the  functions  of  all  angles  whatsoever: 


FUNCTIONS  OF  ANGLES. 


285 


REVOLVING   LINE   IN 

First  quarter, 
Second  quarter, 
Third  quarter, 
Fourth  quarter. 


NE  &  COSEC.       COS.  &  SEC.       TAN.  A  COT. 


+ 
+ 


+ 


+ 


+ 
+ 


836.  Corollary. — The  functions  of  two  angles  are  the 
same,  when  one  of  the  angles  is  greater  than  the  other  by 
four  right  angles. 

EXERCISES. 

827. — 1.  Demonstrate  the  following  equations :  sec.  120°=  —  2; 
COS.  135°=  — V2- 

2.  The  ratio  of  one  straight  line  to  its  projection  upon  another 
is  what  function  of  their  angle  ? 

3.  Construct  an  angle  whose  tangent  is  — 1 ;    one  whose  sine 

is  — ^. 

4.  Construct  an  angle  whose  cosine  is  — f. 


ANGLES    OF    A    GIVEN    FUNCTION. 

828.  Theorem. — Any  given  simple  function,  when  taken 
irrespective  of  its  algebraic  sign,  belongs  to  four  different 
angles  within  each  revolution. 

If  BAG  is  the  acute  angle  of  a  given  function,  the 
revolving  line  AB  will, 
at  some  point  in  each 
quarter  of  its  revolution, 
form  an  acute  angle  with 
XX',  equal  to  the  angle 
BAG.  Now,  the  numer- 
ical value  of  the  function 

depends  upon  the  acute  angle  which  the  revolving  line 
makes   with   the   fixed  line    (817).      Hence,  there   is  an 


286  PLANE  TRIGONOMETRY. 

angle  for  each  quarter  whose  functions  are  numerically 
equal  to  those  of  the  angle  BAG. 

820.  Corollary. — Any  simple  function  of  an  angle  is 
numerically  equal  to  the  same  function  of 

1st.    The  supplement  of  the  angle; 

2nd.  The  given  angle  increased  by  two  right  angles ; 

3rd.    The  given  angle  taken  negatively. 

The  sine  and  cosecant  of  supplementary  angles  have 
the  same  signs,  while  the  other  simple  functions  of  sup- 
plementary angles  have  opposite  signs  (825).  The  cosine 
and  secant  of  an  anoxic  and  of  its  ne";ative  have  the  same 
signs,  while  the  other  simple  functions  of  such  angles  have 
opposite  signs.  The  tangent  and  cotangent  of  an  angle, 
and  of  the  same  angle  increased  by  two  right  angles,  have 
the  same  signs,  while  the  other  simple  functions  of  such 
angles  have  opposite  signs. 

These  conclusions  as  to  the  sine  may  be  expressed 
thus : 

sin.  A=sin.  (180°-A)= -sin.  a80"-|-xV)  =  -sin.  (-A). 

The  following  more  general  expressions  are  easily  de- 
duced from  the  above  corollary.  If  n  is  0,  or  any  integer 
positive  or  negative,  and  A  is  any  angle,  then 

The  formula  7i-180°+(— 1)^A  includes  all  angles  which 
have  the  same  sine  as  A; 

The  formula  n'360°±A  includes  all  the  angles  which 
have  the  same  cosine  as  A;  and 

The  formula  ?i-180°-(-A  includes  all  angles  which  have 
the  same  tangent  as  A. 

830.  Corollary. — Any  simple  function  of  any  angle 
may  be  expressed  in  terms  of  the  same  function  of  an 
acute  auizle. 


FUNCTIONS  OF  ANGLES.  287 


EXERCISES. 

831. — 1.  Make  a  formula  analogous  to  the  above  for  each  of  the 
other  simple  functions. 

2.  Demonstrate  cosec.  600°  =  — f/S;    cot.  405°=  1. 

3.  Write  a  formula  containing  all  the  values  of  A  when  tan. 
A=l. 

LIMITS   OF   FUNCTIONS. 

S32,  Theorem. —  Hie  sine  of  any  angle  can  not  he 
greater  titan  1,  nor  less  than  —1;  and  the  cosine  has  the 
same  limits. 

For  the  leg  of  a  right  angled  triangle  can  not  be 
greater  than  the  hypotenuse;  and,  therefore,  the  sine 
and  cosine  are  fractions  having  the  numerator  less  than 
the  denominator. 

833.  Theorem — The  secant  and  cosecant  can  not  have 
any  values  between  1  and  —  1 ;  and  the  tangent  and  cotan- 
gent  have  no  limits.  ^ 

These  principles  also  follow  immediately  from  the  defi- 
nitions and  the  nature  of  a  right  angled  triangle. 

834.  As  the  revolving  line  passes  through  the  first 
quarter  of  its  revolution,  the  sine  increases  from  0  to  1. 
The  sine  of  a  right  angle  is  unity,  for  in  that  case  the 
perpendicular  coincides  with  the  hypotenuse.  Then  the 
sine  decreases  till  the  angle  is  equal  to  two  right  angles, 
when  the  sine  becomes  0.  It  continues  to  decrease  till 
the  angle  becomes  three  right  angles,  Avhen  the  sine  is 
—  1.  Then  again  it  increases  to  the  end  of  the  revolu- 
tion, where  the  sine  is  0. 

The  cosii.e  of  0°  is  1,  which  decreases  as  the  angle 
increases  till  the  cosine  of  90°  is  0,  and  the  cosine  of 


288  PLANE  TRIGONOMETRY. 

180°  is  —1.  Then  it  increases  through  the  remaining 
half  of  the  revolution. 

The  tangent  of  0°  is  0.  As  the  angle  increases  the 
tangent  increases  without  limit,  and  the  tangent  of  a 
right  angle  is  infinite.  The  tangent  of  an  obtuse  angle 
is  negative,  and  as  the  angle  increases  the  tangent  varies 
from  minus  infinity  to  zero.  In  the  third  quarter  the 
tangent  varies  as  in  the  first  quarter  through  all  possible 
positive  values ;  and  the  variations  of  the  fourth  quarter 
are  like  those  of  the  second. 

The  variations  of  the  cotangent,  secant,  and  cosecant 
may  be  traced  in  the  same  way. 

These  values  of  the  functions  at  particular  points  may 
be  expressed  as  follows : 


0° 

90° 

180° 

270° 

360° 


0  1        0  00        1        00 

1  0  00  0  00  1 
0—10  00—1  00 
1  0  00  0  00—1 
0        1         0  00         1         00 


The  versed  sine  increases  from  0  to  2  as  the  angle 
increases  from  0°  to  180°,  and  decreases  from  2  to  0 
through  the  other  two  quarters. 


EXERCISES. 

835. — 1.  Trace  the  value  of  this  expression:  cos.  A  —  sin.  A. 
as  A  varies  from  0°  to  360°. 

2.  What  are  the  sine  and  the  tangent  of  810°? 

3.  What  are  the  cosine  and  secant  of — 450°? 

4.  What  are  the  cosecant  and  cotangent  of  150°? 

5.  Construct  an  angle  greater  than  90°,  whose  sine  is  ^:    ono 
whose  tangent  is  ^;  one  whose  cosine  is  j,. 


FUNCTIONS  OF  ANGLES.  289 


RELATIONS  BETWEEN  THE  FUNCTIONS. 

836.  A  simple  function  of  an  angle,  being  a  ratio, 
may  be  expressed  as  a  fraction. 

Let  a  be  the  perpendicular,  h  the  base,  and  c  the  hy- 
potenuse of  the  triangle  used  in  defining  the  functions  of 
an  angle.  In  order  to  include  all  possible  angles,  let  it 
be  understood  that  a  and  h  are  either  positive  or  nega- 
tive.    Then, 


sin. 

A=-^ 

C 

COS. 

A=*-, 

c 

tan. 

^'l 

cot. 

-4 

sec. 

'"V 

coscc. 

A=f. 
a 

837.  Corollary — The  sine  and  cosecant  of  air  angle 
are  reciprocals;  also,  the  tangent  and  cotangent  are  re- 
ciprocals; and  the  cosine  and  secant  are  reciprocals. 
That  is, 

sin.  A  cosec.  A  =  1,  tan.  A  cot.  A  =  1,  cos.  A  sec.  A  =  1. 

A  practical  result  of  these  equations  is,  that  the  cose- 
cant, secant,  and  cotangent  are  less  used  than  the  other 
simple  functions.  For,  if  one  has  occasion  to  multiply  or 
divide  by  the  cosecant,  the  object  is  accomplished  by 
dividing  or  multiplying  by  the  sine ;  and  similarly  of  the 
secant  and  cotangent. 

838.  By  means  of  the  Pythagorean  Theorem  and  the 
fractions  just  stated,  any  function  of  an  angle  may  be 
expressed  in  terms  of  any  other  function  of  the  same 
angle.     For  example,  let  it  be  required  to  find  the  value 

Tris— 25. 


290  PLANE  TRIGONOMETRY. 

of  each  of  the  other  simple  functions  in  terms  of  the  sine 
of  the  same  angle.     Beginning  with  the  equation, 

and  dividing  both  members  by  c^, 

That  is,  the  sum  of  the  squares  of  the  sine  and  cosine 
of  any  angle  is  equal  to  unity.     Hence, 

sin.  A  =  Vl  —  cos.^  A ;     also,  cos.  A  =  Vl  —  sin."^  A. 

The  exponent  is  given  to  sin.  and  to  cos.,  because  it  is 
the  function  that  is  involved  and  not  the  angle. 

839.  The  sine  of  an  angle  is  equal  to  the  product  of 

the  tangent  by  the  cosine.     For, 

a       a      h 

-  =  7  X-- 

CDC 

That  is,     sin.  A  =  tan.  A  cos.  A. 

^^  .         sin.  A  sin.  A 

Hence,      tan.  A  = -=    , 

COS.  A      vl  —  sin.'^A 

Since  the  tangent  and  cotangent  are  reciprocals, 

COS.  A       Vl  —  sin.^  A 

cot.  A  =  — r-  = — I • 

sin.  A  sm.  A 

Since  the  secant  and  cosine  are  reciprocals, 


.      ^  .  1  Vsec.^^A  — 1 

sm.   A  =  \  I —r  — r 

^         sec.^A  sec.  A 


FUxNCTiONS  OF  ANGLES.  291 


EXERCISES 


S40. — 1.    By  similar  methods,  find  expressions   for  the  cosine 
and  tangent  in  terms  of  each  of  the  other  functions. 

2.  Render  each  formula  into  ordinary  language.     This  valuable 
exercise  should  be  continued  throughout  the  work. 

3.  Given  2  sin.  A  =  tan.  A,  to  find  A.     Ans.  0°,  60°,  120°,  180°, 
240°,  or  300°. 

4.  If  sin.  A  =  |,  what  is  the  value  of  cos.  A  ? 

5.  If  sin.  A  =  ^,  what  is  the  value  of  tan.  A? 

6.  Demonstrate  sin.\  18°  ^^(/5 — 1).      Notice  that  18°  is  the 
angle  made  by  the  apothegm  and  radius  of  a  regular  decagon. 


FUNCTIONS   OF    (90°  rt  A). 

841.  Theorem. — The  cosine  of  an  angle  is  the  sine  of 
its  complement. 

That  is,  COS.  A  =  sin.  (90° r— A).  For,  in  the  right 
angled  triangle  of  the  definitions,  the  acute  angles  are 
complementary;   and  (818) 

,  COS.  A  =  -  =  sin.  B. 
c 

This  demonstration  appears  to  apply  only  to  the  case 
when  the  angle  A  is  acute,  when  the  revolving  line  is  in 
the  first  quarter.  The  student  may  construct  a  figure  for 
each  of  the  other  quarters,  and  show  that  the  proposition 
is  universally  true. 

842.  Corollary — Similarly,  the  cotangent  and  cose- 
cant are  respectively  the  tangent  and  secant  of  the  com- 
plementary angle.  It  is  from  this  property  that  these 
functions  (cos.,  cot.,  cosec.)  derive  their  names. 


292 


TLA NE  TRIG (JxN ( )M  KTRY. 


843.  Theorem — Sin.  (90''  +  .4)  =  cos.  A,  and  cos. 
(90°  +  A)  =  —  sin.  A. 

It  has  been  proved  that  sin.  A=  sin.  (180°  —  A),  what- 
ever is  the  value  of  A  (829).  It  is  therefore  true  for 
(90°+  A).     Substituting,  we  have 

sin.  (90°+ A)  =sin.  (180°— 90°- A)  =sin.  (90°-A)  =  cos.  A. 

Again,  since  cos.  A=  sin.  (90°  —  A)  for  all  values  of 
A,  then  for  A  we  may  substitute  90°  +  A.     Hence, 

cos.(90°+A)=sin.(90°-90°— A)=sin.(-A)=— sin.A. 


EXERCISES. 

844.— 1.  Find  the  value  of  tan.  (90°  +  A). 

2.  Illustrate  with  diagrams  all  the  principles  of  this  section. 

3.  Given  sin.  A=cos.  2A,  to  find  the  value  of  A. 

|/(10  +  2y/5) 


4.  Demonstrate  tan.  72° 


1/5  — 1. 


FUNCTIONS    OF    TWO    ANGLES. 

845.  Let  the  angle  DCF  be  designated  by  A  and  the 
angle  FCG  by  B;  then  DCG 
is  A-|-B.  From  any  point  G 
in  the  line  CG  let  fall  GH 
and  GF  respectively  perpen- 
dicular to  CD  and  CF.  From 
F  let  foil  FD  and  FK  respect- 
ively perpendicular  to  CD  and 
GIL  Then,  the  angle  FGK  is 
equal  to  FCD,  or  A  (140). 

Now  DF=CFXsin.  A,  and  CF=CGXcos.  B;  hence, 


DF=CGXsin.  A  cos.  B. 


FUN{  TIONS  OF   ANGLES.  293 

Likewise  GK  =  GFXcos.  A,  and   GF=CGXsin.  B; 
hence, 

GK=CGXcos.  A  sin.  B. 

Also,  GK+DF=GK+KH=GH=CGXsin.  (A+B); 
therefore. 


sin 


.  (A+B)=sin.  A  cos.  B-f-cos.  A  sin.  B,       (i.) 


In  the  above  figure  the  given  angles  and  their  sum  are 
acute.  The  same  demonstration  will  apply  for  any  given 
angles,  constructing  the  figure  exactly  according  to  the 
directions,  producing  when  necessary  the  lines  on  which 
the  perpendiculars  fall. 

The  cosine  of  the  sum  of  two  angles  may  be  found  i# 
terms  of  the  sine  and  cosine  of  the  angles,  by  the  above* 
diagram  and  similar  reasoning.     Or,  it  may  be  derived 
from  the  formula  just  demonstrated,  as  follows: 

Regarding  90°  +  A  as  one  angle,  we  have 

sin.(90°+A+B)=sin.(90°+A)cos.B+cos.(90°+A)sin.B. 

Substituting  for  the  functions  of  90°+ A  and  90°+ A+B, 
their  equivalents  (843), 

COS.  (A  +  B)=  cos.  A  COS.  B  —  sin.  A  sin.  B,     (rr.) 

In  these  two  formulas  for  the  sine  and  cosine  of  the 
sum  of  two  angles,  if  —  B  is  substituted  for  B,  then  the 
sign  of  sin.  B  is  changed,  but  not  of  cos.  B  (825).     Thus, 

sin.  (A — B)  =  sin.  A  cos.  B  —  cos.  A  sin.  B,       (iii.) 
COS.  (A  —  B)  =  cos.  A   cos.  B  +  sin.  A  sin.  B,     (iv.) 

These  two  formulas  may  be  demonstrated  independ- 
ently of  the  former,  in  the  same  manner  as  the  formula 
for  the  sine  of  the  sum. 


294  PLANE  TRIGONOMETRY. 

The  tangent  of  the  sum  of  two  angles  is  found  thus : 

. .  j^-r>s  _  sin.  (A-f-B) sin.  A  cos.  B  -\-  cos.  A  sin.  B 

cos.  (A-j-B)     cos.  A  cos.  B  —  sin.  A  sin.  B  ' 

Dividing  both  terms  of  the  fraction  by  cos.  A  cos.  B, 

/A     I    -ON         tan.  A -1- tan.  B  ,    . 

tan.  (A  +  B)  = \ ,     .      (v.) 

^  ^       1  —  tan.  A  tan.  B  ^    ^ 

^.    .,     ,  ,.        .^.  tan.  A — tan.  B  ,      , 

am.lariy,     tan.  (A  -  B)  =  ^^^-^-^-^-^ ,     .     (vi.) 

EXERCISES. 

846. — 1.  Demonstrate  formula  ii  in  the  same  manner  as  for- 
mula I,  and  both  of  them  for  those  cases  where  the  angles  are  not 
acute.     Observe  in  what  quarters  the  sine  and  cosine  are  negative. 

2.  Express  each  formula  in  ordinary  language ;  for  example :  the 
sine  of  the  sum  of  two  angles  is  equal  to  the  sum  of  the  products  of 
the  sine  of  each  by  the  cosine  of  the  other. 

3.  Demonstrate  cos.  12°  =  |-(730  +  6  v/5  +  i/5  —  1.) 

FUNCTIONS  OF  MULTIPLES  AND  PARTS  OF  ANGLES. 

847-  In  the  formulas  of  the  sine,  the  cosine,  and  the 
tangent  of  the  sum  of  two  angles,  suppose  B  =  A ;  then, 

sin.  2A  =  2  sin.  A  cos.  A,  .     .     .     .       (i.) 

COS.  2 A  =  cos.2  A  —  sin/^  A,    .     .     .      (n.) 

2  tan.  A 
tan.2A  =  j_^^^,^,     ....     (m.) 

By  substituting  (n  —  1)A  for  B  in  the  original  formulas, 
sin.  ?^A,  cos.  nA,  and  tan.  nA  may  be  expressed  in  func- 
tions of  A  and  of  (n —  1)A.     Thus,  when  the  functions  of 


FUNCTIONS  OF  ANGLES.  295 

A  are  known,  the  functions  of  2A,  3A,  etc.,  may  be  cal- 
culated. 

Since  cos.^  A  +  sin.^  A  =  1      (838),  we  have 


cos 


.  2A  =  1  —  2  sin.*^  A ;     also,  cos.  2A  =  2  cos.'  A  —  1. 


These  formulas  being  true  for  all  angles,  J  A  may  be 
substituted  for  A.     Then,  transposing, 

2  sin.'  i^  =  1  —  cos.  A,      and  2  cos.'  JA  =  1  +  cos.  A. 

Therefore, 

sin.  iA=  -J I  (1  —  cos.  A), 

COS.  i A  =  Vi(l  +  cos.  A),    ....     (iv.) 

By  these  formulas,  from  the  cosine  of  an  angle,  may 
be  calculated  the  sine  and  cosine  of  its  half,  fourth, 
eighth,  etc. 

EXERCISES. 

«-^      1    Tx  A       sec.  A  —  1 

848, — 1.  Demonstrate  tan. -;r- =  — : : — . 

2  tan.  A 

2.  What  is  the  value  of  sin.  15°;  cos.  3°;  sin.  1°  3CK? 

FORMULAS   FOR  LOGARITHMIC   USE. 

849.  In  order  to  render  a  formula  fit  for  logarithmic 
calculation,  products  and  quotients  must  be  substituted 
for  sums  and  diiferences.  This  may  frequently  be  done 
by  means  of  the  formulas  which  follow. 

The  formulas  for  the  sine  and  cosine  of  (A  ±  B)  be- 
come, by  adding  the  third  to  the  first,  subtracting  the 
third  from  the  first,  adding  the  second  to  the  fourth,  and 
subtracting  the  second  from  the  fourth  (845), 


29G  PLANE  TRIGONOMETRY. 

sin.  (A+B)  +  sin.  (A— B)=2sin.  Acos.B,       (i.) 
sin.  (A  +  B)  —  sin.  (A  —  B)  =  2  cos.  A  sin.  B,      (ii.)     1 

COS.  (A  -|-  B)  +  cos.  (A  —  B)  =  2  cos.  A  cos.  B,  (in.) 

COS.  (A  —  B)  —  cos.  (A  +  B)  ==  2  sin.  A  sin.  B,  (iv.) 

In  the  above,  let  A  -j-  B  =  C,  and  A  —  B  =  D ;  whence, 
A=  i(C  +  D),  and  B=  l(C-D).     Then, 

sin.  C+sin.  D=2  sin.  1{G +T))  cos.  h{^  —  J)),  (v.) 
sin.  C  —  sin.  D  =  2  cos.  1{C  +  D)  sin.  J(C  —  D),  (vi.) 
cos.C+cos.D  =  2  cos.  J(C  +  D)  cos.J(C— D),  (vii.) 
COS.  D  —  COS.  C  =  2  sin.  i{G  +  J))  sin.  |(C  — D),  (viii.) 

By  dividing  v  by  vi, 

sin.  C+sin.  D  wn  i  t\\      *.  ^ /n     t\\     tan.  J(C+D) 

-, — -J—, — -  =  tan.  J  (C+D)  cot.  i(C— D)= ^)^  '  ^^;. 

sm.C— sin.D  ^v     i      /         2\  i     tan.  i(C— D) 

Hence, 
sin.  C+sin.D  :  sin. C— sin.  D  :  :  tan.  K^+D)  :  tan.  i(C— D).     (ix.) 

EXERCISES. 

850. — 1.  Demonstrate  sin.  5A=:5sin.  A  —  20sin.'A+16  sin.^A. 
2.  Demonstrate  sin.  (A  +  B)  sin.  (A  —  B)  =  sin.^A  —  sin.-^B. 

TRIGONOMETRICAL   TABLES. 

851.  By  the  application  of  algebra  to  the  geometrical 
principles  used  in  the  construction  of  regular  polygons, 
the  student  has  found  that  the  sine  of  30°  is  J,  and  the 
sine  of  18°  is  ^(-^5 —  1).     From  these  may  be  found  the 


TRIGONOMETRICAL  TABLES.  297 

cosines  of  these  angles ;  then  (847,  iv)  the  sine  and  co- 
sine of  15°,  and  then  the  sine  of  3°  (845,  in).  The  sine 
of  1°  may  be  found  as  follows : 

sin.  3A=sin.  (A+2A)  =  sin.  Acos.2A+  cos.  A  sin.  2A. 

Substituting  the  values  of  cos.  2 A  and  sin.  2 A  (847), 

sin.  3A  =  3  cos.^  A  sin.  A  —  sin.^  A. 

Hence  (838),  sin.  3A  =  3  sin.  A  —  4  sin.^  A. 

Put  1°  for  A ;  then,  knowing  the  value  of  sin.  3°,  and 
representing  the  unknown  sin.  1°  by  x, 

Only  one  of  the  roots  of  this  equation  is  less  than  sin. 
3°.  It  must  be  sin.  1°,  and  may  be  calculated  by  alge- 
braic methods  to  any  required  degree  of  approximation. 

Similarly,  an  equation  of  the  fifth  degree,  may  be 
formed  from  the  value  of  sin.  5A ;  and  by  its  means  from 
the  known  sin.  1°  may  be  found  sin.  12'.  Thus,  by  suc- 
cessive steps,  the  functions  of  1'  and  of  V  may  be  found 
to  any  required  degree  of  accuracy. 

Having  the  sine  and  cosine  of  these  small  angles,  the 
functions  of  their  multiples  may  be  calculated  (847).  This 
method,  however,  is  tedious  and  is  not  used  in  practice. 
It  serves  to  show  the  possibility  of  calculating  these  func- 
tions by  elementary  algebra  and  geometry.  The  higher 
analysis  teaches  briefer  methods. 

These  numerical  functions  are  called  the  natural  sines, 
tangents,  etc.,  to  distinguish  them  from  the  logarithmic 
functions  which  will  be  defined  presently. 


•29S  PLANE  TRIGONOMETRY. 

852.  The  Table  of  Natural  Sines  and  Tangents 
gives  these  functions  to  six  places  of  figures  for  every  10' 
from  0  to  90°.  It  also  serves  as  a  table  of  cosines  and 
cotangents. 

If  the  sine  or  tangent  of  some  intermediate  angle  is 
required,  it  may  be  found  by  taking  a  proportional  part 
of  the  difference,  with  as  much  accuracy  as  the  functions 
given  in  the  table,  except  when  the  angle  is  nearly  a  right 
angle.  For  example,  to  find  the  sine  84°  23'  30'',  the 
table  gives  the  sine  of  34°  20' =.564007.  Since  3'  30" 
is  .35  of  10',  multiply  2399,  the  difference  between  this 
sine  and  that  of  34°  30',  by  .35,  and  add  the  product  to 
the  given  sine;  the  sum  .564847  is  the  natural  sine  of 
30°  23'  30". 

At  the  beginning  of  this  table,  the  functions  vary  with 
almost  perfect  uniformity,  and  in  proportion  to  the  angle. 
Thus,  the  sine  and  the  tangent  of  100'  differ  only  by  one- 
millionth  from  one  hundred  times  the  sine  or  the  tangent 
of  1'.  At  the  close  of  the  table,  the  tangent  varies  rap- 
idly and  the  sine  varies  slowly,  and  both  irregularly. 
Therefore,  for  the  intermediate  angles  (those  not  given 
in  the  table),  the  last  lines  are  less  to  be  rehed  upon 
than  the  first. 

The  tangent  of  a  large  angle  may  be  found  with  greater 
accuracy  by  finding  the  cotangent  of  the  same  angle  and 
taking  its  reciprocal  (837). 

LOGARITHMIC   FUNCTIONS. 

S53.  Before  proceeding  to  the  study  of  this  article, 
the  student  should  understand  the  use  of  the  tables  of 
loiirarithms  of  numbers. 

A  logarithmic  sine,  tangent,  etc.,  means  the  logarithm 
of  the  sine,  of  the  tangent,  etc.     In  the  tables,  the  char- 


TRIGONOMETRICAL  TABLES.  299 

acteristic  of  every  logarithmic  trigonometric  function  is 
increased  by  10.  For  example,  sin.  30°  =  I ;  log. 
1  =  1.698970,  which  is  the  true  logarithm  of  the  sine  of 
30°;  but  the  tabular  logarithmic  sine  of  30°  is  9.698970. 
The  object  of  this  arrangement  is  simply  to  avoid  the 
use  of  negative  characteristics,  as  would  be  the  case  with 
all  the  sines  and  cosines  and  half  of  the  tangents  and  co- 
tangents. Therefore,  whenever  in  a  calculation,  a  tabu- 
lar logarithmic  function  is  added,  10  must  be  subtracted 
from  the  result  to  find  the  true  logarithm;  and  whenever 
a  tabular  logarithmic  function  is  subtracted,  10  must  be 
added  to  the  result.  If,  however,  in  place  of  subtracting 
a  logarithmic  function,  the  arithmetical  complement  is 
added,  the  result  does  not  need  correction,  the  10  to  be 
added  for  one  reason,  balancing  that  to  be  subtracted  for 
the  other. 

854.  The  table  gives  the  logarithmic  sine,  tangent, 
cosine,  and  cotangent  for  every  1'  from  0  to  90°.  The 
degrees  are  marked  at  the  top  of  each  page  and  the  min- 
utes in  the  left  hand  column  descending,  for  the  sines  and 
tangents ;  and  the  degrees  at  the  bottom  of  each  page 
and  the  minutes  in  the  right  hand  column  ascending,  for 
the  cosines  and  cotangents.  The  columns  marked  P.  P.  1'' 
contain  the  proportional  part  for  one  second,  to  facilitate 
the  proper  addition  or  subtraction. 

In  using  the  proportional  part  for  the  cosine  and  co- 
tangent, remember  that  these  functions  decrease  when  the 
angle  increases. 

855,  To  find  the  logarithmic  sine,  etc.,  of  a  given  angle. 

If  the  angle  is  expressed  in  degrees  only,  or  in  degrees 
and  minutes,  take  the  corresponding  sine  or  other  function 
directly  from  Table  lY. 

If  the  angle  is  expressed  in  degrees,  minutes,  and  sec- 


300  PLANE  TRIGONOMETRY. 

onds,  then  take  the  logarithmic  function  corresponding 
to  the  given  degrees  and  minutes;  multiply  the  propor- 
tional part  for  V^  by  the  number  of  seconds;  and  ndd  the 
product  to  the  tabular  function,  for  the  sine  and  tangent, 
and  subtract  it  for  the  cosine  and  cotangent. 

For  example,  to  find  the  tabular  logarithmic  sine  of 
40°  13'  14''     . 

tab.  log.  sin.  40°  13'  =  9.810017, 
P.P.I"  =2.5,    ...    2.5x14     ..     =  35, 

Therefore,   .     .    tab.  log.  sin.  40°  13'  14"  =  9.810052. 

To  find  the  tabular  logarithmic  cosine  of  75°  40'  21'^ 

tab.  log.  COS.  75°  40'  =  9.393685, 
P.  P.  1"  =  8.23,    .     .    8.23  X  21     .     .     =  173, 

Therefore,  .     .    tab.  log.  cos.  75°  40'  21"  =  9.393512. 

This  method  of  using  the  proportional  part  given  in 
the  tables,  gives  results  that  are  true  to  six  decimal 
places,  except  for  the  sines,  tangents,  and  cotangents  of 
angles  less  than  three  degrees,  and  for  the  cosines  and 
cotangents  of  angles  greater  than  eighty-seven  degrees. 

The  sines  and  tanojents  of  small  anojles  increase  almost 
uniformly.  Therefore,  the  logarithmic  sine  and  tangent 
of  one  of  these  small  angles  may  be  found  nearly,  by 
adding  to  the  logarithmic  sine  or  tangent  of  one  second 
the  logarithm  of  the  number  of  seconds  in  the  given 
angle.  This  result  is  subject  to  the  correction  in 
Table  V. 

The  cosines  and  cotangents  of  large  angles  are  found 
in  the  same  way,  since  they  are  the  sines  and  tangents 
of  the  small  angles  (841  and  842.) 

Since  the  tangent  and  cotangent  of  an  angle  are  recip- 
rocals, the  rule  just  given  for  finding  the  tangents  of  small 


TRIGONOMETRICAL  TABLES.  301 

angles,  may  be  applied  to  the  cotangents  also.     For  the 

correction,  see  Table  V. 

For  example,  to  find  the  logarithmic  sine  of  45'  23''  = 

2723", 

add  to     .      4.685575, 
log.  2723,      3.435048, 

8.120623. 
Subtract  as  in  Table  V,  13, 

tab.  log.  sin.  45' 23"    =  8.120610. 

856.  To  find  the.  angle  when  its  logarithmic  sine,  tan- 
gent, cosine,  or  cotangent  is  given. 

If  the  given  function  is  found  in  Table  IV,  take  the 
corresponding  angle,  expressed  in  degrees,  or  in  degrees 
end  minutes. 

If  the  given  function  is  not  in  the  table,  take  that 
which  is  next  less;  subtract  it  fi-om  the  given  function; 
divide  the  remainder  by  the  proportional  part  for  1" ;  the 
quotient  is  the  number  of  seconds,  to  be  added,  in  case 
of  sine  or  tangent,  to  the  angle  corresponding  to  the 
tabular  function  used ;  and  to  be  subtracted  in  case  of 
the  cosine  or  cotangent. 

For  example,  to  find  the  angle  whose  tabular  logarith- 
mic tangent  is 10.456789,  ^ 

tab.  log.  tan.  70°  44'  =  10.456501, 

P.P.I"  =6.75,      ....  288-^6.75  =  43. 

Therefore,  70°  44'  43"  is  the  angle  sought. 

To  find  the  angle  whose  tabular  logarithmic  cotan- 
gent is 9.876543, 

tab.  log.  cot.  53°  3'  =  9.876326, 

P.  P.  1"=  4.38,     .     .     .     .  217^4.38=50. 


302  PLANE  TRIGONOMETRY. 

Therefore,  53°  2'  10''  is  the  angle  Avhose  logarithmic 
cotanirent  is  9.876543. 

o 

When  great  accuracy  is  desired  and  the  angle  to  be 
found  is  less  than  three  degrees  or  greater  than  eighty- 
seven,  the  corrections  in  Table  V  may  be  used,  first  using 
Table  IV  to  determine  the  angle  approximately. 

RIGHT    ANGLED    TRIANGLES. 

857.  The  principles  have  now  been  established,  by 
which,  whenever  certain  parts  of  a  triangle  are  knowri, 
the  remaining  parts  can  be  calculated.  Since  the  trig- 
onometrical functions  are  the  ratios  between  the  sides  of 
a  right  angled  triangle,  the  problems  concerning  such 
triangles  need  no  other  demonstration  than  is  contained 
in  the  definitions. 

The  sum  of  the  acute  angles  being  90°,  when  one  is 
known,  the  other  is  found  by  subtraction. 

858.  Problem — Given  the  hypotenuse  and  one  angle, 
to  find  the  other  parts. 

The  product  of  the  hypotenuse  by  the  sine  of  either 
acute  angle,  is  the  side  opposite  that  angle.  The  prod- 
uct of  the  hypotenuse  by  the  cosine  of  either  acute 
angle,  is  the  side  adjacent  to  that  angle. 

859.  Problem. —  Given  one  leg  and  one  angle,  to  find 
the  other  parts. 

The  quotient  of  one  leg  divided  by  the  sine  of  tlie 
opposite  angle  is  the  hypotenuse.  The  product  of  oi.e 
leg  by  the  tangent  of  the  adjacent  angle  is  the  other  leg. 

860.  Problem. — Given  one  leg  and  the  hypotenuse,  to 
find  the  other  parts. 

The  quotient  of  one  leg  divided  by  the  hypotenuse  is 


RIGHT  ANGLED  TRIANGLES.  303 

the  sine  of  the  angle  opposite  that  leg,  and  the  cosine  of 
the  adjacent  angle.  The  other  leg  may  then  be  found 
by  the  previous  problem. 

861.  Problem — Given  the  two  legsi  to  find  the  other 

parts. 

The  quotient  of  one  leg  divided  by  the  other  is  the 
tangent  of  the  angle  opposite  the  dividend.  The  hypot- 
enuse may  then  be  found  by  the  second  problem. 

When,  as  in  the  last  two  problems,  two  sides  are  given, 
the  third  may  be  found  by  the  Pythagorean  Theorem. 

862.  Only  the  sine,  cosine,  and  tangent  are  used  in 
the  above  solutions.  The  student  may  easily  propose 
solutions  by  means  of  the  other  functions.  Since  none 
of  the  above  problems  requires  addition  or  subtraction, 
the  operations  may  all  be  performed  by  logarithms. 

For  example  :  A  railroad  track,  463  feet  3  inches  long, 
has  a  uniform  grade  of  3°.  How  high  is  one  end  above 
the  other?  Here  the  hypotenuse  and  one  acute  angle  are 
given,  to  find  the  opposite  side. 

log.  463.25         =  2.665815, 
tab.  log.  sin.  3°=  8.718800, 

Omittinor  the  tabular  10,  the  sum  1.384615  is  the 
logarithm  of  24.2446.  Hence,  the  ascent  is  nearly  24 
feet  3  inches. 

EXERCISES. 

863. — 1.  Construct  a  figure  to  illustrate  the  above,  and  each  of 
the  following. 

2.  The  hypotenuse  is  4321,  one  angle  is  25°  30^.  Find  the  other 
angle  and  the  two  legs.  Solve  this  both  with  and  without  loga- 
rithms. 


•504  PLANE  TRIGONOMETRY. 

3.  Two  posts  on  the  bank  of  a  river  are  one  hundred  feet  apart ; 
the  line  joining  them  is  perpendicular  to  the  line  from  the  first 
post  to  a  certain  point  on  the  opposite  bank;  and  the  same  lino 
makes  an  angle  of  78°  52^  with  the  line  from  the  second  post  to  the 
same  point  on  the  opposite  bank.     How  wide  is  the  river? 

4.  The  instrument  used  in  measuring  the  angle  in  the  above 
statement  is  imperfect,  the  observations  being  liable  to  an  error  of 
V.     To  what  extent  does  that  affect  the  calculated  result  ? 

5.  The  hypotenuse  being  7093,  and  one  leg  2308.5,  find  the  other 
leg  and  the  angles. 

6.  An  observer  standing  60  feet  from  a  wall  measures  its  angu- 
lar height,  and  finds  it  to  be  15°  37^,  his  eye  being  5  feet  from  the 
ground,  which  is  level.     How  high  is  the  wall? 

7.  How  much  would  the  last  result  be  affected  by  an  error  of  5^^ 
in  observing  the  angle? 

8.  How  much  if  there  had  also  been  an  error  of  2  inches  in 
measuring  the  horizontal  line  ? 

9.  Find  the  apothegm  and  radius  of  a  regular  polygon  of  7  sides, 
one  side  being  10  inches. 

10.  Find  the  area  of  a  regular  dodecagon,  the  side  being  2  feet. 

11.  The  legs  being  42.9  and  47.52,  find  the  angles  and  the  hy- 
potenuse. 

12.  A  tower  103  feet  high  throws  a  shadow  51.5  feet  long  upon 
the  level  plane;  what  is  the  angle  of  elevation  of  tlie  sun  r* 

13.  How  much  would  the  last  result  be  affected  by  an  error  of 
3  inches  in  the  given  height  or  length  ? 


SOLUTION  OF  PLANE  TRIANGLES. 

864.  One  angle  of  a  triangle  being  the  supplement  of 
the  sum  of  the  other  two,  when  two  are  known  the  third 
may  be  found  by  subtraction.  Also,  the  sine  bf  either 
angle  is  equal  to  the  sine  of  the  sum  of  the  other  two. 

The  letters  a,  b,  and  c  represent  the  sides  of  a  triangle 
respectively  opposite  the  angles  A,  B,  and  C. 


PLANE  TRIANGLES. 


305 


865.  Theorem. — The  square  of  one  side  of  a  triangle 

is  equal  to  the  sum  of  the  squares  of  the  other  two  sides, 
less  twice  the  product  of  those  sides  by  the  cosine  of  their 
included  angle. 

For,  in  the  first  figure  (411), 
a-'=b''+e'  —  2b'A'D, 
and  in  the  second  figure  (412), 

a'=b'+c'+2b'AD; 
but  in  the  first  case, 
AD  =  COS.  A  X  AB  =  c  cos.  A ; 
and  in  the  second, 

AD  =  —  cos.  A  X  AB  =  —  c  cos.  A. 

Substituting  these  values  of  AD  in  their  respective 
equations,  both  become 

a-=  b--\-  e~ —  2bc  cos.  A. 

By  similar  reasoning,  it  may  be  shown  that 

b^  =  a^  -{-  c^  —  2ac  cos.  B, 
and  c^=:  a^+  b^—  2ab  cos.  C. 

These  three  equations  suffice  for  the  solution  of  all 
problems  on  plane  triangles,  but  they  are  not  suitable  for 
logarithmic  calculations.  The  following  are  not  liable  to 
this  objection : 

TriiT.— 26. 


30G  PLANE  TRIGONOMETRY. 

866.   Theorem — Expressing  the  sum  of  the  sides  of 
any  triangle  hy  p,  then  sin.  —  =  ^^^ lAM-. ^. 

For,  by  the  formula  just  demonstrated, 

COS.  A  = !-— . 

2bc 
Hence  (847,  iv). 


.n4  =  ,/,<r.rwA)  =  J.0_-::^l=^), 


=  V AT. =  V 


sin.  ^  -  ,  K--b'+  2be  -e^_Jip-  26)  (p  -  2c) 


4bc  »  4bc 


4=V^ 


2       ^  be 

Similarly,  find  sin.  JB  and  sin.  JC  in  terms  of  the  sides. 

The  cosine  and  the  tangent  may  also  be  expressed  in 
terms  of  the  sides,  as  follows:     By  Art.  847,  iv, 


A 

cos 


4-vw+^^)-M'  +  '^,^)- 


Reducing,    cos.  — -=a/^ 


^-JhP  ihp-^) 


2        ^  he 


Also,  tan.  ^  =  ^^  =  JMEAME^, 

2       cos.  JA      >/       yp  {yp  -  a) 

Similarly,  find  the  cosine  and  tangent  of  JB  and  of  JC. 


PLANE  TRIANGLES. 


307 


867.  Theorem — The  sides  of  any  triangle  are  pro- 
portional  to  the  sines  of  the  opposite  angles. 

That  is,   a  :h  ::  sin.  A  :  sin.  B. 

For,  whether  A  is    acute    or 
obtuse, 

BD  =  ABsin.  A, 

and         BD  =  BC-sin.  C. 

Therefore,  c  sin.  A  =  a  sin.  C, 

and  a  :  c  ::  sin.  A  :  sin.  C. 

Similarly,  a  :  b  ::  sin.  A  :  sin.  B. 

868.  Theorem. — One  side  of  a  triangle  is  equal  to  the 
sum  of  the  products  found  by  multiplying  each  of  the  other 
sides  by  the  cosine  of  the  angle  which  it  forms  with  the  first 
side. 

For,      AC  =  CD  ±  DA  =  BC-cos.  C  +  BA-cos.  A. 
That  is,     b  =  a  cos.  C  -\-  c  cos.  A. 

869.  Theorem. —  The  sum  of  any  two  sides  of  a  tri- 
angle is  to  their  difference  as  the  tangent  of  half  the  sum 
■of  the  tivo  opposite  angles  is  to  the  tangent  of  half  their 

difference. 

By  Art.  867,         a  :  b  ::  sin.  A  :  sin.  B. 
By  composition  and  division, 

a-{-b  :  a  —  b  ::  sin.  A  +  sin.  B  :  sin.  A  —  sin.  B. 
Hence  (849,  ix), 

a+b  :  a  —  b  ::  tan.  J(A.  +  B)  :  tan.  J  (A  —  B.) 


308  PLANE  TRIGONOMETRY. 

8T0.  Problem. — Given  the  sides  of  a  triangle,  to  find 
the  angles. 

This  rule  is  derived  from  the  formula  for  the  sine  of 
half  an  angle  (866). 

From  half  the  sum  of  the  sides,  subtract  each  of  the 
sides  adjacent  to  the  required  angle ;  multiply  together 
these  remainders;  divide  this  product  by  the  product  of 
the  two  adjacent  sides,  and  extract  the  square  root  of  the 
quotient.     This  root  is  the  sine  of  half  the  angle  sought. 

The  student  may  write  rules  for  the  solution  of  this 
problam  from  the  formulas  for  cos.  JA,  and  tan.  JA,  and 
cos.  A. 

For  example,  the  given  sides  are  a  =  3457,  b  =  4209, 
and  c  =  6030.4.  For  finding  all  the  angles,  the  formula 
for  the  tangent  of  half  an  angle  is  the  best,  because  the 
same  numbers  are  used  for  every  angle.  To  find  the 
angle  C, 

i^p         =6848.2  Jj? -5  =2639.2 

.i.p_«  =  3391.2  lp  —  c=    817.8 

log.  (Ip  —  a)  .  =  3.530353 
log.  ihp—b)  .  =  3.421472 
a.cAog.  Ip  .  .  =  6.164424 
a.cAog.  {yp—c)=    7.087353 

tab.  log.  tan.2  JC  =  20.203602 
tab.  log.  tan.   JC  =  10.101801, ' 

which  is  the  tab.  log.  tan.  51°  39'  16'^4.     Therefore,  the 
angle  C  is  103°  18'  33''. 

In  the  above  calculation,  the  sum  of  the  logarithms 
exceeds  by  20  the  sum  required,  on  account  of  the 
arithmetical  complement  twice  used ;  but  the  tabular 
logarithm  of  tan.^  JC  being  also  20  more  than  the  true 
logarithm  of  tan.'-*  .^C,  no  correction  is  necessary. 


PLANE  TRIANGLES.  309 

Find  in  a  similar  manner  the  other  two  angles,  and  test 
the  result  by  comparing  the  sum  with  180°. 

There  is  another  method  of  solving  this  problem.  By 
dividing  any  triangle  into  two  right  angled  triangles,  if 
the  sides  are  known,  the  altitude  and  the  segments  of  the 
base  may  be  found  (328).  Then  the  angles  may  be  cal- 
culated as  in  the  solutions  of  right  angled  triangles. 

8T1.  Problem — Given  two  angles  and  a  side,  to  find 
the  other  angle  and  sides. 

Find  the  third  angle  by  subtracting  the  sum  of  the 
given  two  from  180°.  Then  find  the  remaining  sides  by 
the  formula  (867), 

sin.  A  :  sin.  B  : :  a  :  J. 

8T2.  Problem. — Given  two  sides  and  an  angle  opposite 
one  of  them,  to  find  the  other  angles  and  side. 

Find  the  angle  opposite  the  other  given  side  by  the 
formula, 

a  :  b  :  :  sin.  A  :  sin.  B. 

Find  the  third  angle  by  subtraction,  and  the  third  side 
by  the  formula, 

sin.  A  :  sin.  C  :  :  a  :  e. 

When  the  side  opposite  the  given  angle  is  equal  to  or 
greater  than  the  other  given  side,  there  can  be  only  one 
solution  (287).  When  it  is  less  than  the  other  given 
side,  there  may  be  two  solutions  (291  and  300).  This  is' 
called  the  ambiguous  case.  The  result  is  indicated  by 
the  trigonometrical  formula,  for  the  angle  is  found  by 
its  sine  ;  and  for  a  given  sine  there  are  two  angles,  one 
acute  and  one  obtuse. 


310  ■    PLANE  TRIGONOMETRY. 

The  side  opposite  the  given  angle  may  be  so  small  as 
to  make  the  triangle  impossible  (300.)  This  result  is 
also  indicated  by  the  trigonometrical  solution,  for  the  sine 
of  the  angle  sought  is  found  to  be  greater  than  unity, 
which  is  impossible. 

873.   Problem — Given    two    sides   and    the   included 

angle,  to  find  the  othef'  angles  and  side. 

Find  the  sum  of  the  other  angles  by  subtraction,  and 
the  difference  of  those  angles  by  the  formula  (869), 

a  +  b  :  a  —  b  ::  tan.  i(A+B)  :  tan.  i(A  — B). 

Knowing  half  the  sum  and  half  the  difference  of  the 
two  required  angles,  take  the  sum  of  these  two  quantities 
for  the  greater  and  their  difference  for  the  less  of  the 
angles.  The  third  side  is  found  as  in  the  preceding 
problems. 

This  problem  may  be  solved,  without  logarithms,  by 
the  formula  (865), 

a^  =b^^  c^  —  2bc  COS.  A. 


AREAS. 

874.  Theorem — The  area  of  a  triangle  is  equal  to  half 
the  product  of  any  two  sides  multiplied  by  the  sine  of  the 
included  angle. 

Thus,  the  area  of  triangle  ABC  =  ^  be  sin.  A. 

For  the  altitude  BD  (see  last  figure)  is  the  product  of 
the  side  c  by  the  sine  of  the  angle  A. 

The  student  may  now  review  Art.  390. 


PLANE  TRIANGLES. 


311 


Take  some  point  C,  from  which 


APPLICATIONS. 

875. 1.   To  measure  the  distance  from  one  point  to 

another,  when  the  line  between  them  can  not  be  passed  over 
with  the  measuring  chain  or  rod. 

Let  A  and  B  be  the  two  points, 
both  A  and  B  are  visible,  and 
such  that  the  lines  AC  and  BC 
can  be  measured  with  the  rod  or 
chain.  Measure  these  and  the 
angle  C.  Then,  in  the  triangle 
ABC,  two  sides  and  the  included 

angle  are  known,  from  which  the  third  side  AB  can  be  calculated 
(873). 

If  A  and  B  are  visible  from  each  other,  as  when  the  obstacle 
between  them  is  open  Water^  then  the  angles  A  and  B  may  be  ob- 
served. In  that  case  it  is  necessary  to  measure  only  one  of  the 
sides  AC  or  BC ;  for,  knowing 
sides  may  be  calculated  (871) 


one  side  and  the  angles,  the  other 


2.  To  find  the  height  and  distance  of  an  inaccessible 
object. 

Let  P  be  the  top  of  all  object,  whose  distance  from,  and  height 
above,  the  point  A  are  required.  At 
A  observe  the  angle  PAC,  that  is, 
the  angle  of  inclination  of  the  line 
AP  with  the  plane  of  the  horizon 
(537  and  563).  Then,  measure  any 
length  AB,  on  a  horizontal  line 
directly  towards  the  object,  and  at 
B  observe  the  angle  PBC. 

In  the  triangle  APB,  the  side  AB  and  the  angle  A  are  known  ; 
also  the  angle  ABP,  since  it  is  the  supplement  of  PBC;  hence,  AP 
can  be  calculated.  Then  PC  =  AP-sin.  A,  and  AC  =  AP-cos.  A; 
thus  determining  the  height  and  distance  of  the  object. 

The  angle  A  is  called  the  angular  elevation  of  the  point  P  as 
seen  at  A,  the  angle  PBC  being  the  elevation  of  the  same  point 
as  seen  at  B.  If  P  were  below  the  level  of  A,  the  angle  thus  ob- 
served would  be  the  angular  depressioyi  of  the  object. 


312 


PLANE  TRIGONOMETRY 


When,  as  is  generally  the  case,  it  is  inconvenient  to  measure 
the  line  AB  "on  a  horizontal  line 
directly  toward  the  object,"  meas- 
ure any  length  AB  in  any  conve- 
nient direction;  at  A,  observe  the 
angle  PAB,  and  the  elevation  PAC; 
and  at  B,  observe  the  angle  PBA. 
Then,  in  the  triangle  APB,  the  side 
AB  and  the  adjacent  angles  being 
known,  the  side  AP  may  be  found, 
and  the  height  and  distance  of  P 
calculated  as  before. 

3.  To  find  the  distance  between  two  visible  but  inaccessi- 
ble objects. 

Let  P  and  N  be  the  objects,  C  and  B  two  accessible  points  from 
vrhich  both  the  objects  are  visi- 
ble. At  C  observe  the  angles 
PCN  and  NCB,  and  if  C,  B,  N,  P 
are  not  all  in  the  same  plane,  ob- 
serve also  the  angle  PCB.  At 
B  observe  the  angles  PBC  and 
NBC.     Measure  CB. 

In  the  triangle  PCB,  the  side 
CB  and  its  adjacent  angles  being 
known,  the  side  CP  can  be  found. 

In  the  triangle  NCB,  the  side  CB  and  its  adjacent  angles  being 
known,  the  side  CN  may  be  found.  Then,  in  the  triangle  PCN, 
the  sides  CP  and  CN  and  their  included  angle  being  known,  the 
side  PN  may  be  found. 

4.  To  find  the  ividth  of  a  river  without  an  instrument 
for  observing  angles. 

Let  P  be  a  visible  point  on  the  further  bank,  and  A  a  point 
opposite  to  it  on  this  side.     Take 

B,  C,  and  D,  any  convenient  ac- 
cessible points,  such  that  B,  A, 
and  P  are  in  a  straight  line,  and 

C,  D,  and  P  are  in  a  straight  line ; 
and  measure  AB,  AC,  AD,  BD, 
and  CD. 


PLANE  TRIANGLES.  313 

All  the  sides  of  the  triangles  ABD  and  ACD  being  known,  the 
angles  BAD  and  ADC  may  be  found,  and  hence  their  supplements 
DAP  and  ADP.  Then,  from  the  side  AD  and  the  two  adjacent 
angles  of  the  triangle  ADP,  the  side  AP  may  be  calculated. 


EXERCISES. 

876. — 1.  The  sides  of  a  triangle  being  70,  80,  and  100,  what 
are  the  angles? 

2.  Two  angles  of  a  triangle  are  76°  30^  23^^  and  54°  IV  5V', 
and  the  side  opposite  the  latter  is  40.451.;  find  the  other  sides. 

3.  Two  sides  of  a  triangle  are  243.775  and  907.961,  and  the 
angle  opposite  the  former  is  15°  16^  17^^;  find  the  other  parts. 

4.  Two  sides  of  a  triangle  are  196.96  and  173.215,  and  the  in- 
cluded angle  40°;  find  the  other  angles  and  side. 

5.  From  a  station  B,  at  the  base  of  a  mountain,  its  summit  A  ia 
seen  at  an  elevation  of  60°;  after  walking  one  mile  towards  the 
summit,  up  a  plane,  making  an  angle  of  30°  with  the  horizon  to 
another  station  C,  the  angle  BCA  is  observed  to  be  135°.  Find  the 
height  of  the  mountain. 

6.  Two  sides  of  a  parallelogram  are  25  and  17.101,  and  one  of 
its  diagonals  38.302;  find  the  other  diagonal. 

7.  A  person  observing  the  elevation  of  a  spire  to  be  35°,  advances 
80  yards  nearer  to  it,  and  then  finds  the  elevation  is  70° ;  required 
the  height  of  the  spire. 

8.  From  the  top  of  a  tower  whose  height  is  124  feet,  the  angles 
of  depression  of  two  objects,  lying  in  the  same  horizontal  plane 
with  the  base  of  the  tower  and  in 
48°;  what  is  their  distance  apart? 


Trio:. —27, 


314  SPHERICAL  TRIGONOMETRY 


CHAPTER    XIII. 
SPHERICAL    TRIGONOMETRY. 

877.  Spherical  Trigonometry  is  the  investigation 
of  the  relations  which  exist  between  the  sides  and  angles 
of  spherical  triangles. 

Each  side  of  a  spherical  triangle  being  an  arc,  is  the 
measure  of  an  angle.  It  has  the  same  ratio  to  the  whole 
circumference  that  its  angle  has  to  four  right  angles.  It 
may  be  measured  by  degrees,  minutes,  and  seconds,  as 
an  angle  is  measured.  It  has  its  sine,  tangent,  and  other 
trigonometrical  functions  ;  it  being  understood  that  the 
sine,  etc.,  of  an  arc  are  the  sine,  etc.,  of  the  angle  at  the 
center  which  that  arc  subtends. 

The  propositions  Avhich  express  the  relations  between 
the  sides  and  angles  of  a  spherical  triangle,  apply  equally 
well  to  the  faces  and  diedral  angles  of  a  triedral  (766 
and  seq.).  If  the  investigation  were  made  from  this  point 
of  view,  as  it  well  might  be,  the  proper  title  of  the  subject 
would  be  Trigonometry  in  Space. 

THREE    SIDES    AND   AN    ANGLE. 

878.  Theorem — The  cosine  of  any  side  of  a  spherical 
triangle  is  equal  to  the  product  of  the  cosines  of  the  other 
two  sides,  increased  by  the  product  of  the  sines  of  those 
sides  and  the  cosine  of  their  included  angle. 


SPHERICAL  ARCS  AND  ANGLES. 


315 


Let  ABC  be  a  spherical  triangle,  0  the  center  of  the 
sphere,  AD  and 
AE  tangents  re- 
spectively to  the 
arcs  AB  and  AC. 
Thus,  the  angle 
EAD  is  the  angle 
A  of  the  spherical  0 
triangle ;  the  angle 
EOD  is  measured 
by  the  side  a,  and 
so  on. 

From  the  triangles  EOD  and  EAD   (865), 

DE'=  0D'+  OE  —  20D0E  cos.  a, 
DE'=  AD'+ AE  —  2AD- AE  cos.  A. 

By  subtraction,  the  triangles   OAE   and   OAD  being 
right  angled, 

0  =  20T+  2AD-AE  cos.  A  —  20D-0E  cos.  a; 


rru      r  OA   OA  ,   AE   AD  . 

Therefore,       cos.  a  =  -^  .  ^^+  -^--  .  ^  cos.  A; 


that  is. 


cos.  a  =  COS.  b  COS.  c  +  sin.  h  sin.  e  cos.  A. 


In  the  above  construction,  the  sides  which  contain  the 
angle  A  are  supposed  less  than  quadrants,  since  the  tan- 
gents at  A  meet  OB  and  OC  produced.  That  the  for- 
mula just  demonstrated  is  true  when  these  sides  are  not 
less  than  quadrants,  is  shown  thus : 


316 


SPHERICAL  TRIGONOMETRY. 


Suppose  one  of  the 
sides  greater  than  a 
quadrant,  for  example, 
AB.  Produce  BA  and 
BC  to  B',  and  repre- 
sent AB^  and  CB'  by  d 
and  a'  respectively. 

Then,  in  the  triangle  AB'C,  as  just  demonstrated, 

cos.  a'=  cos.  h  cos.  c'  -\-  sin.  h  sin.  c'  cos.  B'AC. 

Now,  a',  c\  and  B'AC  are  respectively  supplements  of 
a,  c,  and  BAG.     Hence, 

cos.  a  =  COS.  h  cos.  c  -\-  sin.  b  sin.  c  cos.  A. 

When  botJh  the  sides  -which  contain  the  angle  A  arc 
greater  than  quadrants,  produce  them  to  form  the  aux- 
iliary triangle,  and  the  demonstration  is  similar  to  the 
last. 

Suppose  that  one  of  the  sides  b  and  c  is  a  quadrant, 
for  example,  e.  On  AC, 
produced  if  necessary,  take 
AD  equal  to  a  quadrant, 
and  join  BD.  Now  A  is  a 
pole  of  the  arc  BD  (754), 
and  therefore  that  arc 
measures  the  angle  A 
(760). 

Then,  from  the  triangle  BCD, 

cos*  a  =  cos.  CD  cos.  BD  +  sin.  CD  sin.  BD  cos.  CDB ; 

but  —  CD    is   the  complement   of   b,  BD   measures  A, 


SPHERICAL  ARCS  AND  ANGLES.  317 

and  CDB   is  a   right  angle.     Hence,  this   equation  be- 
comes, 

COS.  a  =  sin.  h  cos.  A, 

and  the   formula  to    be   demonstrated   reduces   to   this, 
when  c  is  a  quadrant. 

The  proposition  having  been  demonstrated  for  any 
angle  of  any  spherical  triangle, 

cos.  h  =  COS.  a  COS.  c+  sin.  a  sin.  c  cos.  B, 
cos.  c  =  COS.  a  COS.  h  +  sin.  a  sin.  h  cos.  C. 

These  have  been  called  the  fundamental  equations  of 
Spherical  Trigonometry.  By  their  aid,  when  any  three 
of  the  elements  of  a  spherical  triangle  are  known,  the 
others  may  be  calculated. 

A    SIDE    AND    THE    THREE    ANGLES. 

879.  Since  the  formulas  just  demonstrated  are  true 
of  all  spherical  triangles,  they  apply  to  the  polar  triangle 
of  any  given  triangle.  Therefore,  denoting  the  sides  and 
angles  of  the  polar  triangle,  by  accenting  the  letters  of 
their  corresponding  parts  in  the  given  triangle, 

COS.  a'  =  cos.  h'  cos.  c'  -\-  sin.  h'  sin.  c'  cos.  A', 

but  a'=180°  — A,  5^=180°  — B,  and  A'  =  180°— «, 
etc.  (777).     Substituting  these  values  of  a',  h',  etc., 

COS.  (180°—  A)  =  COS.  (180°  — B)  cos.  (180°— C)  + 

sin.  (180°  — B)  sin.  (180°— C)  cos.  (180° -«). 


318  SPHERICAL  TRIGONOMETRY. 

Reducing  (829),  and  changing  the  signs, 

COS.  A  =  —  COS.  B  COS.  C  +  sin.  B  sin.  C  cos.  a. 

Similarly, 

cos.  B  =  —  cos.  A  COS.  C  +  sin.  A  sin.  C  cos.  6, 
cos.  C  =  —  COS.  A  cos.  B  +  sin.  A  sin.  B  cos.  e. 

None  of  the  above  formulas  is  suited  for  logarithmic 
calculation. 

FORMULAS    FOR    LOGARITHMIC    USE. 

880.   Let  p  represent  the  perimeter,  that  is,  p  == 
a+b+c. 

By  transposing  and  dividing  the  fundamental  formula 

(878), 

.  COS.  a cos.  6  cos.  C  rT^^  o  fOAr  \ 

cos.  A  =  : — Therefore  (845,  iv), 

sin.  0  sin.  c 


^  .     sin.6  sin.c+cos.5  cos.c — cos.a    cos.(6— c)— cos.a 

1— cos.A= ;    ^    . = ;^ — -A 

sm.osin.  c.  sm.  osin.c. 

Substituting  for  this  numerator  its  value  (849,  viii), 
and  dividing  by  2, 

J-(l  -  COS.  A)  =  si"-  K^+  ^  —  0)  sin.  ),{a-h-\-c) 
^  sin.  b  sin.  c. 

Substituting  p  for  its  value,  and  extracting  the  root 
(847,  IV), 


Sin 


■^  _     pin.  (ij?  —  6)sin.  (Vp  —  c) 
2        ^  sin.  h  sin.  c. 


SPHERICAL  ARCS  AND  ANGLES.  319 

To  find  tbe  value  of  the  cosine  of  half  the  angle, 

sin. 6  sin.c?— C0S.6  cos.c-fcos.a    cos. a — cos.(6+6') 


l-fcos.A=z- 


s'm.b  sin.  c.  sin.  b  sin.  c. 


Hence,       cos.^  =  >""  ^-^  f -^^^'"^ 
2        ^  sin.  6  sin.  c. 

Dividing  sin.  ^  A  by  cos.  J  A, 


tan.  ^  =  J^^'^l^""  ^^  ^'"'  ^^P~''^ 

"  ^         sin    1- 


Q 


sin.  Ip  sin.  (Ip  —  a) 


Find  the  analogous  formulas  for  the  sine,  cosine,  and 
tangent  of  JB  and  of  JC. 

881.  Let  E  represent  the  spherical  excess,  that  is, 
E=A  +  B  +  C-180°. 

By  reasoning  upon  the  polar  triangle  as  in  the  pre- 
ceding article,  the  formula  for  the  sine  of  half  an  angle 
becomes 

180°— a_^  /sin.  A(lcS()°— A+B— C)  sin.  ^(180°— A— B+C)^ 
sin.  (180'— B)  sin.  (180°  — C) 


r—a  Isin.  I 


but       sin.  i(180°—  a)  =  sin.  (90°—  »«)  =  cos.  },a, 
and      sin.  J(180°  — A +  B  — C)  =  sin.  (B  -  JE),  etc. 


Therefore,     cos.  ^  ^  J^^"' (^- ^^)  ^'"- ^^- ^^) . 
2        ^^  sin.  B  sin.  C 

Similarly,  from  the  formula  for  the  cosine  of  half  the 


angle. 


sin  -  =  Jsin.|Esin.(A-^-E) 
'2       ^Z        sin.  B  sin.  C 


320 


SPHERECAL  TRIGONOMJ]TRY. 


Hence,       tan. 


sin.  JE  sin.  (A  —  JE) 


\sin.  (B— IE)  sm.  (C  — ^E)  * 


Since  E  must  be  less  than  360°  (771),  sin.  JE  is  pos- 
itive; and  since  sin.  ^a  is  a  real  quantity,  sin.  (A —  ?,E) 
must  be  positive.  Therefore,  any  angle  of  a  spherical 
triangle  is  greater  than  half  the  spherical  excess. 


OPPOSITE    SIDES    AND    ANGLES. 

882.  Theorem — The  sines  of  the  angles  of  a  spherical 
triangle  are  proportional  to  the  sines  of  the  opposite  sides. 

Let  ABC  be  the  spherical  triangle,  and  0  the  center 
of  the  sphere.  From  any 
point  P  in  OA,  let  PD  fall 
perpendicular  to  the  plane 
BOC;  make  DE,  DF  per- 
pendicular respectively  to 
BO,  OC;  and  join  PE,  PF, 
and  OD. 

The  plane  PED  is  per- 
pendicular to  the  plane  BOC 
(556).  Therefore,  OE  is 
perpendicular  to  the  plane  PED,  the  angle  PED  is  the 
same  as  the  angle  B  (759),  and  PEO  is  a  right  angle. 
Therefore,  PE  =  OP  •  sin.  POE  =  OP  •  sin.  c;  and 
PD  =  PE  •  sin.  B  =  OP  •  sin.  c  sin.  B. 

Similarly,      PD  =  OP  •  sin.  b  sin.  C ; 

therefore,  OP  *  sin.  c  sin.  B  =  OP  •  sin.  6  sin.  C. 

sin.  B       sin.  b 


sin.  C       sin.  e 


or     sin.  B  :  sin.  C  : :  sin.  b  :  sin.  c 


SPHERICAL  ARCS  AND  ANGLES.      s  321 

The  figure  supposes  h,  c,  B,  and  C  to  be  each  less  than 
90°.  When  this  is  not  the  case,  the  figure  and  the  dem- 
onstration are  slightly  modified.  For  example,  when  B 
is  greater  than  a  right  angle,  the  point  D  falls  beyond 
BO,  and  PED  becomes  the  supplement  of  B,  having  the 
same  sine. 


FOUR    CONTIGUOUS    PARTS. 

883.  Theorem — The  product  of  the  cotangent  of  one 
side  hy  the  sine  of  another^  is  equal  to  the  product  of  the 
cosine  of  the  included  angle  by  the  cosine  of  the  second 
side,  plus  the  product  of  the  sine  of  the  included  angle  hy 
the  cotangent  of  the  angle  opposite  the  first  side. 

We  have  (878  and  882), 

COS.  a  =  COS.  h  COS.  c+  sin.  h  sin.  c  cos.  A, 
cos.  c  ==.  COS.  a  cos.  b  -}-  sin.  a  sin.  b  cos.  C, 

sin.  a  sin.  C 


sm.  c  = 


sin.  A 


Eliminate  c  by  substituting  these  values   of  cos.  c  and 
sin.  c  in  the  first  equation, 

,  ,   ,     .         .     ,         ^,,         ,    ,  sin.asin.icoa.Asin.C 

COS. a  =  (cos.a  C0S.6  +  sin.a  sin.6  cos.C )  cos.o  H -. -. ; 

^  ^  sm.  A 

transposing  and  reducing,  since  1  —  cos.^5  =  sin.^  6, 

cos.asin.^6=sin.a  sin.^cos.i  cos.C+sin.a  sin.^cot.Asin.C ; 

dividing  by  sin.  a  sin.  b, 

cot.  a  sin.  b  =  cos.  b  cos.  C  +  cot.  A  sin.  C. 


322  SPHERICAL  TRTGONOMETRY. 

The  demonstration  being  general,  may  be  applied  to 
other  angles  and  sides,  making  these  five  additional 
formulas : 

cot.  h  sin.  a  =  cos.  a  cos.  C  +  cot.  B  sin.  C, 
cot.  h  sin.  c  =  cos.  c  cos.  A  +  cot.  B  sin.  A, 
cot.  c  sin.  h  =  cos.  b  cos.  A+  cot.  C  sin.  A, 
cot.  e  sin.  a  =  cos.  a  cos.  B  +  cot.  C  sin.  B, 
cot.  a  sin.  c  =  cos.  e  cos.  B  -|-  cot.  A  sin.  B. 

FORMULAS    OF    DELAMBRE. 

884.  Putting  J  A  and  JB  for  A  and  B  respectively,  in 
formula  I,  Art.  845, 

sin.  J  (A  +  B)  =  sin.  JA  cos.  JB  +  cos.  ^A  sin.  ^B. 

Substitute  the  values  of  the  factors  of  the  second  mem- 
ber, as  found  in  Art.  880, 

.     A+B      sin.(Jp— a)+sin.(Jjt?— 5)    k'm.lp  sin.  {}^p~c)  ^ 

sm.  — - —  = ; \—      :  ;     J       ; 

2  sm.  c  '       sm.  a  sm.  o 

but, 
.  N      .    /,        IN        .     /^      a—h\      .     Ic      a—h\ 

(849,1),    ....   =2sin.  ic  cos.  |(a_5), 
and  (847,  i),      sin.  c=2sin.  Jc  cos.  ^c. 

Substituting  these  values,  also  cos.  \Q>  for  the  radi- 
cal (880), 

.     A  +  B      cos.  Ua—h)  ,^ 

sm. ^- —  = -^ COS.  ?,C, 

2  cos.-Jc 


sm 


or. 


sin.  J(A  +  B)  _  COS.  l{a—h) 

COS.  JC  COS.  \c 


SPHERICAL  ARCS  AND  ANGLES.  323 

Similarly,  by  beginning  with  formulas  ii,  iii,  and  iv 
of  Art.  845,  we  find, 


sin.  |(A  — B)    _  sin,  ^(a  —  h) 
COS.  JC        ~~        sin.  ^c 

COS.  |(A  +  B)  _  COS.  Ija  +  h) 
sin.  JC  COS.  Ic 

cos.|(A  —  B)       sin.  ^(a+  5) 
sin.  JC  sin.  \c 

These  four  formulas  of  Delambre  were  published  by 
him  in  1807. 

NAPIER'S    ANALOGIES. 

885.  Divide  the  first  of  the  formulas  of  Delambre  by 
the  third,  the  second  by  the  fourth,  then  the  fourth  by 
the  third,  and  the  second  by  the  first,  and  these  results 
are  obtained: 

tan.  |(A  +  B)       cos.  l(a  —  b) 
cot.  JC  COS.  2(^+  ^) 

tan.  |(A  —  B)  _  sin.  » (a  —b) 
cot.  AC  sin.  i(a  +  5) 

tan.  l{a+b)  __  cos.  i(A  —  B) 
tan.  ic       ~"  COS.  i(A  +  B) ' 

tan.  ^(a  —  b)  __  sin.  |(A  —  B) 
tan.  |c        """  sin.  J(A  +  B) 

These  formulas  may  be  stated  as  proportions,  and  are 
called  Napier's  Analogies,  from  their  inventor,  analogy 
being  formerly  used  as  synonymous  with  proportion. 


324  SPHERICAL  TRIGONOMETRY. 

886.  In  the  first  of  the  above  equations,  cos.  l(a  —  h) 
and  cot.  |C  are  necessarily  positive;  hence,  tan.  J(A-j-  B) 
and  COS.  J(«+  b)  are  of  the  same  sign;  thus,  ^(A+B) 
and  li^"^^)  ^^^  either  both  less  or  both  greater  than 
ninety  degrees. 

In  the  second  of  the  above  equations,  sin.  Jl'^  ~)~  ^) 
and  cot.  jC  are  positive;  hence,  tan.  J(A  —  B)  and  sin. 
l{a  —  b)  have  the  same  sign  ;  thus,  ^(A — B)  and  l(a  —  h) 
are  either  both  positive,  both  negative,  or  both  zero. 
Therefore,  in  any  spherical  triangle,  the  greater  angle 
is  opposite  the  greater  side,  and  conversely. 

EXERCISES. 

8Sf. — 1.  Find  the  formula  that  results  from  applying  the  prin- 
ciple of  polar  triangles  to  the  first  of  Napier's  Analogies ;  also,  to 
the  first  formula  of  Art.  883. 

2.  State  a  theorem  applying  the  principle  of  Art.  878  to  triedrals. 

3.  Show,  from  the  third  of  Napier's  Analogies,  that  the  sum  of 
any  two  sides  of  a  spherical  triangle  is  greater  than  the  third. 

EIGHT  ANGLED  SPHERICAL  TRIANGLES. 

888.  The  foregoing  formulas  may  be  applied  to  right 
angled  triangles  by  supposing  one  of  the  angles  to  be 
right,  for  example  A.      In  this  manner  we  have : 

Art.  878,  1st  formula,  cos.  a  =  cos.  b  cos.  e,  .      (i.) 

Art.  879,  1st  formula,  cos.  a  =  cot.  B  cot.  C,  .    (ii.) 

Art.  882,  sin.  b  =  sin.  a  sin.  B  1  /      v. 

"       "  sin.  a  =  sin.  a  sin.  C  ^ 

Art.  883,  1st  formula,  tan.  b  =  tan.  a  cos.  CI  /     >. 

"       "       6th  formula,  tan.  c  =  tan.  a  cos.  B  J 


RIGHT  ANGLED  TRIANGLES.  325 

Art.  883,  3rd  formula,  tan.  6  =  sin.  c  tan.  B  1         ,    s 

"       "  4th  formula,  tan,  c  =  sin.  b  tan.  C  J 

Art.  879,  2nd  formula,  cos.  B  =  sin.  C  cos.  b\       ,     . 

"       ^'  3rd  formula,  cos.  C  =  sin.  B  cos.  c  j 

In  deducing  li,  iv,  and  V,  the  formulas  are  reduced 
somewhat  by  divisions.  These  are  sufficient  for  the  so- 
lution of  every  case.  These  principles  may  be  stated  as 
follows : 

cos.  hyp.  =  product  of  cosines  of  sides, 
cos.  hyp.  =  product  of  cotangents  of  angles, 
sine  side    =  sine  opposite  angle  X  sine  hyp., 
tan.  side    =  tan.  hyp.  X  cosine  included  angle, 
tan.  side    =  tan.  opposite  angle  X  sine  other  side, 
cos.  angle  =  cos.  opposite  side  X  sine  other  angle. 

889.  Since  the  cosine  of  the  hypotenuse  has  the  same 
sign  as  the  product  of  the  cosines  of  the  other  two  sides, 
it  follows  either  that  two  of  these  three  cosines  are  neg- 
ative, or  none.  Therefore,  in  a  right  angled  spherical 
triangle,  either  all  the  sides  are  less  than  quadrants,  or 
two  are  greater  and  one  is  less. 

It  appears  also  (v)  that  the  tangent  of  an  oblique  an- 
gle and  of  its  opposite  side  have  the  same  sign.  There- 
fore, these  two  parts  of  the  triangle  are  either  both  less 
or  both  greater  than  90°.  This  is  expressed  by  saying 
they  are  of  the  same  species. 


NAPIER'S  RULE  OP  CIRCULAR  PARTS. 

800.  A  mnemonic  rule  for  the  formulas  of  right  angled 
spherical  triangles  was  invented  by  Napier,  and  published 
with  his  description  of  logarithms  in  1614. 


326  SPHERICAL  TRIG0N0MI:T11Y. 

The  right  angle  being  omitted,  five  parts  of  the  triangle 
remain.  The  two  sides  which  include  the  right  angle, 
the  complements  of  the  other  angles,  and  the  complement 
of  the  hypotenuse  are  called  the  circular  parts  of  the 
triangle.  These  are  supposed  to  be  arranged  around  a 
circle  in  the  order  they  occur  in  the  triangle.  Any  one 
of  the  five  circular  parts  may  be  called  the  middle  part, 
then  the  two  next  to  it  are  the  adjacent  parts,  and  the 
remaining  two  are  the  opposite  parts. 

Napier's  rule  is :  The  sine  of  the  middle  part  is  equal 
to  the  product  of  the  tangents  of  the  adjacent  parts, 
also  to  the  product  of  the  cosines  of  the  opposite  parts. 

The  words  sine  and  middle  having  their  first  vowel  the 
same,  also  the  words  tangent  and  adjacent,  also  the 
words  cosine  and  opposite,  renders  this  rule  very  easy  to 
remember.  For  example,  if  the  complement  of  the  hy- 
potenuse be  the  middle  part,  then  the  complements  of  the 
angles  are  the  adjacent  parts,  and  the  sides  are  the  op- 
posite parts ;  this  gives  formulas  I  and  ii. 

SOLUTION  OF  EIGHT  ANGLED  TRIANGLES. 

891.  Problem — Given  the  hypotenuse  and  an  ojblique 
angle^  to  find  the  other  angle  and  the  sides. 

Find  the  other  oblique  angle  by  formula  ii,  the  side 
opposite  the  given  angle  by  iii,  and  the  adjacent  side 
by  IV. 

For  example,  given  the  hypotenuse  64°  17'  35'',  and 
an  angle  70°,  to  find  the  opposite  side, 

tab.  log.  sin.  70°    .     .     =  9.972986, 
tab.  log.  sin.  64°  17'  35"  =  9.954737, 

tab.  102.  sin.  57°  51'  11"  =  9.927723. 


RIGHT  ANGLED  TRIANGLES.  327 

Therefore,  the  required  side  is  57°  51'  11^'.  It  is 
known  to  be  acute  because   its  opposite  angle  is  acute 

(889). 

892.  Problem — Given  one  side  ayid  the  adjacent  ob- 
lique angle,  to  find  the  other  sides  and  angle. 

Find  the  hypotenuse  by  IV,  the  other  side  by  v,  and 
the  otlier  angle  by  vi. 

893.  Problem — Given  the  two  sides,  to  find  the  hy- 
potenuse and  angles. 

Find  the  hypotenuse  by  I,  and  the  angles  by  v. 

894.  Problem — Given  the  hypotenuse  and  one  side,  to 
find  the  angles  and  the  other  side. 

Find  the  included  angle  by  iv,  the  other  side  by  I,  and 
the  remaining  angle  by  III. 

895.  Problem — Given  the  tivo  oblique  angles,  to  find 
the  three  sides. 

Find  the  hypotenuse  by  li,  and  the  other  sides  by  VI. 

In  the  above  solutions  there  is  no  ambiguous  case. 
Whenever  a  part  is  found  by  means  of  its  sine,  its  spe- 
cies is  determined  by  the  principle  of  Art.  889.  In  the 
1st  and  4th  problems,  if  the  given  parts  are  both  of  90°, 
the  triangle  is  indeterminate.  The  student  may  show 
why. 

896.  Problem Give7i  a  side  and  its  opposite  angle, 

to  find  the  other  sides  and  angle. 

Find  the  hypotenuse  by  ill,  the  other  side  by  V,  and 
the  other  nngle  by  vi. 


328 


SPHERICAL  TRIGONOMETRY. 


Here  the  triangle  is  ambiguous,  as  all  the  parts  are 
found  by  their  sines.  Sup- 
pose BAG  to  be  a  triangle 
right  angled  at  A,  and  that 
C  and  c  are  the  given  parts. 
Produce   CB  and  CA  to 
meet  in  C.     Then  the  tri- 
angle CAB  has  the  same  conditions  as  the  given  triangle, 
for  it  has  a  rigtit  angle  at  A,  the  given  side  BA,  and  C'  = 
C,  the  given  angle. 

897.  The  solution  of  an  oblique  triangle  may  be  made 
in  some  cases  to  depend  immediately  upon  the  solution  of 
a  right  angled  triangle.  If  a  triangle  has  one  of  its  sides 
a  quadrant,  then  its  polar  triangle  has  its  corresponding 
angle  a  right  angle.  The  polar  triangle  can  be  solved  by 
the  preceding  methods,  and  thus  the  elements  of  the  prim- 
itive triangle  become  known. 

If  a  triangle  is  isosceles,  an  arc  from  the  vertex  to  the 
middle  point  of  the  base  divides  it  into  two  equal  right 
angled  triangles,  by  the  solution  of  which  the  elements 
of  the  isosceles  triangle  are  found. 

If  a  triangle  has  two  sides  supplementary,  as  b  and  c, 
the  sides  a  and  c  may  be 
produced  to  B',  making 
the  isosceles  triangle 
B'AC,  which  may  be 
solved  as  above,  giving 
the  elements  of  the  orig- 
inal triangle. 

If  a  triangle  has  two  of  its  angles  supplementary,  then 
its  polar  triangle  has  two  of  its  sides  supplementary. 
This  may  be  studied  in  the  manner  just  stated,  and  thus 
the  parts  of  the  primitive  triangle  become  known. 


SPHERICAL  TRIANGLES.  329 


EXERCISES. 

898. — 1.  Show  that  in  a  right  angled  spherical  triangle,  a  side 
is  less  than  its  opposite  angle  when  both  are  acute,  and  greater 
when  both  are  obtuse. 

2.  The  sides  are  57°  5F  8^^  and  35°  23^  30^^;  find  the  hypotenuse 
and  the  angles. 

3.  The  hypotenuse  is  71°  W  ZT'  and  one  angle  79°  56^  4^^;  find 
the  sides  and  the  other  angle. 

4.  One  side  is  140°,  the  opposite  angle  is  138°  14^  14^^;  find  the 
remaining  parts. 

5.  Show  that  if  the  hypotenuse  is  90°,  one  of  the  sides  must  bo 
90°,  and  conversely. 

6.  The  sides  are  90°,  76°  49'  55^',  41°  45M6'';  find  the  angles. 

7.  A  lateral  edge  of  a  pyramid  whose  base  is  a  square,  makes 
angles  of  60°  and  65°  respectively  with  the  two  conterminous  sides 
of  the  base ;  find  the  diedral  angle  of  that  edge. 


SOLUTION  OF  SPHERICAL  TRIANGLES. 

899.  Problem. — Given  the  sides,  to  find  the  angles. 

Either  of  the  angles  may  be  found  by  the  formulas  of 
Art.  880.  When  all  the  angles  are  required,  the  formula 
for  the  tangent  is  to  be  preferred. 

900.  Problem. — Giveri  the  angles,  to  find  the  sides. 

Either  of  the  sides  may  be  found  by  the  formulas  of 
Art.  881. 

901.  Problem — Given  two  sides  and  the  included 
angle,  to  find  the  other  angles  and  side. 

The  half  sum  of  the  other  angles  may  be  found  by  the 
first  of  Napier's  Analogies,  and  the  half  difference  by  the 

Triir._28. 


330  SPHERICAL  TRIGONOMETRY. 

second;  and  hence,  the  angles  themselves.  Then  the 
third  side  may  be  found  by  the  proportion  of  Art.  882. 
If  the  ambiguity  attendant  upon  the  use  of  the  sine  is 
not  removed  by  observing  that  the  greater  side  of  a  tri- 
angle is  always  opposite  the  greater  angle  (886),  then 
the  third  side  may  be  found  by  Art,  881,  or  by  the  third 
or  fourth  of  Napier's  Analogies,  or  by  one  of  the  formu- 
las of  Delambre. 

For  example,   given   the   side  a  =  76°  35'  36'',  h  = 
50°  10'  80",  and  the  angle  C  =  34°  15'  3". 

By  the  1st  analogy, 

tan.KA+B)=cot.^c;;;;^g-g. 

tab.  log.  cot.  i-C  .  .  .  =  10.511272 
tab.  log.  cos.  J(a  —  6)  .  =  9.988355 
a.  c.  tab.  log.  cos.  \{a-\-h)  =  0.348717 
tab.  log.  tan.  |(A  +  B)  =  10.848344 
.  • .  J(A  +  B)        =  81°  55'  47" 

By  the  2nd  analogy, 

^sin.  l(a  —  h) 
tan.  KA  — B)  =  cot.  ?,C^. f,     ,      '• 

tab.  log.  cot.  JC  .  .  .  =  10.511272 
tab.  log.  sin.  J  (a  — ^^)  .  =  9.358899 
a,  e.  tab.  log.  sin.  l{a+h)  =  0.048648 
tab.  log.  tan.  i(A—  B)  =  9.918819 
.•.J(A— B)       =39°  40' 38" 

Hence,  A  =  121°  86'  20", 

and    *  B  =  42°  15'  14". 


SPHERICAL  TRIANGLES.  331 

Since  the  remaining  side  must  be  less  than  either  of 
the  given  sides,  it  may  be  found  by  the  proportion, 

sin.  A  :  sin.  C  : :  sin.  a  :  sin.  c; 

or  by  the  4th  analogy,  as  follows : 

sin.  J(A+B) 


tan.  3(?=tan.  |(a  —  h) 


sin.  i(A-B) 


tab.  log.  tan.  J(a  —  6)  .  =  9.370544 
tab.  log.  sin.  |(A  +  B)  .  =  9.995677 
a,  c,  tab.  log.  sin.  » (A  —  B)  =  .194877 
tab.  log.  tan.  Jc     .     .     .     =  9.561098 


002.  Problem — Given  one  side  and  the  adjacent 
angles,  to  find  the  other  sides  and  angle. 

The  half  sum  of  the  other  sides  may  be  found  by  the 
3rd  analogy,  and  the  half  difference  by  the  4th;  and 
hence,  the  sides  themselves.  Then  the  third  angle  may 
be  found  by  the  proportion  of  Art.  882. 

If  the  ambiguity  attendant  upon  the  use  of  the  sine  is 
not  removed  by  observing  that  the  greater  angle  is  op- 
posite the  greater  side,  then  it  may  be  found  by  Art.  880, 
or  by  the  1st  or  2nd  analogy,  or  by  one  of  the  formulas 
of  Delambre. 

003.  Problem — Given  two  sides  and  an  angle  opposite 
one  of  them,  to  find  the  other  angles  and  side. 

The  angle  opposite  the  other  given  side  may  be  found 
by  Art.  882,  and  then  the  remaining  angle  and  side  from 
Napier's  Analogies. 

Since  the  sine  is  used  to  find  the  first  angle,  there  may 
be  two  solutions.     The  ambiguity  is  sometimes  removed 


332  SPHERICAL  TRIGONOMETRY. 

by  observing  that  the  greater  angle  is  opposite  the  greater 
side.  When  only  one  value  of  the  angle  found  from  its 
sine  is  consistent  with  this  principle,  there  is  but  one 
solution. 

When  both  values  of  the  angle  thus  found  are  consist- 
ent with  this  principle,  there  are  two  solutions,  that  is, 
there  are  two  distinct  spherical  triangles  which  have  the 
given  elements.  When  the 
angle  A  and  the  sides  a  and 
h  are  given,  h  being  greater 
than  <2,  if  both  values  found 
for  B  are  greater  than  A, 
then  there  are  two  triangles, 
ABC  and  AB'C,  which  have 
the  given  sides  and  angle. 

When  the  same  parts  are  given,  and  h  is  less  than  a, 
if  both  values  found  for  B  are  less  than  A,  there  are 
two  solutions.  In  this  case  the  given  angle  must  have 
been  obtuse,  and  in  the  former  case  it  must  have  been 
acute. 

It  may  happen  that  neither  value  of  the  angle  found 
from  its  sine  is  consistent  with  the  principle  stated.  This 
shows  that  the  given  conditions  are  incompatible,  and  thf.t 
the  triangle  is  impossible. 

OOir.  Problem. — Given  two  ayigles  and  a  side  opposite 
one  of  them,  to  find  the  other  sides  and  angle. 

The  side  opposite  the  other  given  angle  may  be  found 
by  the  proportion  of  Art.  882,  and  then  the  remaining 
angle  and  side  from  Napier's  Analogies,  as  in  the  pre- 
ceding solution. 

This  case  is  precisely  analogous  to  the  last;  it  pre- 
sents the  same  ambiguity,  and  the  ambiguity  is  resolved 
in  the  same  manner. 


SPHERICAL  TRIANGLES.  ^33 


EXERCISES. 

905.— 1.  The  sides  are  60°  4^  54^^,  135°  49^  2(r^,  and  146°  SV 
\y^]  find  the  angles. 

2.  Find  the  diedral  angle  of  a  regular  tetraedron. 

3.  The  sides  are  105°,  90°,  and  75°;  find  the  sines  of  the  angles 
without  the  use  of  the  tables. 

4.  The  angles  are  32°  26^  V,  36°  45^  28^^  and  130°  y  23^^;  find 
the  three  sides. 

5.  Two  sides  are  70°  and  80°,  and  the  included  angle  130°;  find 
the  remaining  angles  and  side. 

6.  Two  sides  are  89°  16^  54^^  and  52°  39^  y,  the  angle  opposite 
the  former  is  70°  39^;  find  the  remaining  parts. 

7.  Given  the  latitude  of  Paris  48°  50^  12'^,  the  latitude  of  New 
York  40°  42^  43^^,  and  the  longitude  of  New  York  west  of  Paris 
76°  20^  27^^,  to  find  the  distance  between  these  points,  along  an  arc 
of  a  great  circle;  the  earth  being  considered  a  sphere  of  a  radius 
of  3956  miles. 

8.  How  much  would  the  last  result  be  affected  by  an  error  of 
2^^  in  the  given  longitude  ?  in  one  of  the  given  latitudes  ? 


334  TRlGONOxMETRY. 


CHAPTER    XIV. 

LOGARITHMS. 

906.  Nearly  all  trigonometrical  calculations  are  made 
by  means  of  logarithms.  To  understand  this  chapter, 
the  student  must  be  acquainted  with  the  algebraic  theory 
of  positive  and  negative  exponents.  He  may  refer  to  the 
algebra  for  an  investigation  of  the  principles  and  the 
methods  of  calculating  tables. 

COMMON    LOGARITHMS. 

907.  The  CoxMMON  Logarithm  of  a  number  is  the 
exponent  of  that  power  of  10  which  is  equal  to  the  num- 
ber.    Hence, 

The  logarithm  of      10  is  1, 

"  "  "  1000   "  3,  etc. 

Again,      the  logarithm  of  1  is      0, 

U  ii 


ii  ii 


"     Jo  or    .1   "  -1, 

"  xJoOr.Ol  "-2,  etc. 


Numbers  greater  than  unity  have  positive  logarithms; 
numbers  less  than  unity  have  negative  logarithms.  The 
powers  of  10  have  the  positive  integers  for  their  log- 
arithms, and  the  reciprocals  of  those   powers   have  the 


LOGARITHMS.  335 

negative  integers  for  their  logarithms.  No  other  num- 
bers have  integral  logarithms.  That  part  of  a  logarithm 
which  is  not  integral  is  always  expressed  by  decimals. 

CHARACTERISTIC. 

908.  The  Characteristic  of  a  logarithm  is  its  in- 
tegral part. 

The  Mantissa  of  a  logarithm  is  the  decimal  part. 

For  convenience  of  calculation,  it  is  an  established 
rule  that  the  mantissa  of  a  logarithm  is  always  positive, 
and  only  the  characteristic  of  a  negative  logarithm  is 
negative.  To  express  this,  the  negative  sign  is  written 
over  the  characteristic.     Thus, 

log.    .2  =  1.301080  =  —  1  +  .301030, 
log  .08  =  2.903090  =  —  2  +  .908090. 

If  any  number  is  between  1  and  10,  its  logarithm  is 
between  0  and  1 ;  if  a  number  is  between  10  and  100,  its 
logarithm  is  between  1  and  2,  and  so  on ;  the  character- 
istic of  the  logarithm  is  always  one  less  than  the  number 
of  integral  places  in  the  given  number.  If  the  number 
is  between  1  and  .1,  its  logarithm  is  between  0  and  — 1  ; 
hence,  its  characteristic  is  — 1.  If  the  number  is  be- 
tween .1  and  .01,  its  logarithm  is  between  — 1  and  — 2; 
hence,  its  characteristic  is  — 2,  and  so  on.  The  charac- 
teristic of  the  logarithm  of  a  fraction  is  numerically  one 
more  than  the  number  of  ciphers  between  the  decimal 
point  and  the  first  significant  figure  of  the  given  fraction 
written  decimally. 

The  student  who  has  learned  the  theory  of  algebraic 
signs  will  perceive  that  the  above  rules  are  included  in 
the  following: 


336  TRIGONOMETRY. 

The  characteristic  of  the  logarithm  denotes  hoiv  many 
places  the  first  significant  figure  of  the  number  is  to  the 
left  of  the  unit's  place. 

The  characteristics  of  logarithms  are  not  given  in  the 
tables,  but  must  be  found  as  above.  If  this  rule  be  taken 
conversely,  it  shows  how  to  place  the  decimal  point,  when 
the  number  is  found  from  its  given  logarithm. 

TABLE    OF    LOGARITHMS. 

909.  Let  c  represent  the  characteristic  and  d  the 
mantissa  of  any  logarithm,  and  let  N  represent  the 
number. 

By  the  definition,       10^+^      =  N. 

Multiplying  by  10,     10«  +  i-J-^=  ION. 

That  is,  if  (?-}"  ^  ^s  th®  logarithm  of  N,  c-f-  1  +  ^  is 
the  logarithm  of  ION,  the  mantissa  of  each  being  d. 
Hence,  multiplying  a  number  by  10  does  not  change  the 
mantissa  of  its  logarithm,  and  it  is  the  same  when  the 
number  is  muLiplied  or  divided  by  any  power  of  10.  In 
other  words :  if  two  numbers  have  the  same  significant 
figures,  their  logarithms  have  the  same  mantissas. 

For  example, 

log.        5  =    .698970, 

log.  5000  =  3.698970, 

log.  .005  =  3.69897C. 

The  table  in  this  work  gives  the  mantissa  of  tlie  log- 
arithm of  every  number  from  1000  to  11000.     It  follows 


LOG  Alii  iHMS.  337 

that  the  mantissa  of  the  logarithm  of  every  number  less 
than  11000  may  be  found  in  the  table. 

The  first  three  or  four  figures  of  each  number  are 
given  in  the  left  hand  column  (see  Table);  the  next 
figure,  at  the  head  and  at  the  foot  of  the  several  columns 
of  mantissas.  The  mantissas  in  the  column  under  0  are 
given  to  six  decimal  places.  The  first  and  second  deci- 
mal figures  of  this  column  are  understood  to  be  repeated 
across  the  page,  and  for  the  spaces  in  the  lines  below. 
In  the  remaining  columns,  1  to  9,  only  the  last  four  of 
the  six  decimal  figures  of  each  mantissa  are  given. 

When  the  second  decimal  figure  changes  from  9  to  0, 
the  remaining  mantissas  of  the  line  are  marked,  to  indi- 
cate that,  in  these  cases,  the  first  two  decimal  figures  are 
taken  from  the  line  below. 

The  last  column  contains  the  difference  between  two 
successive  mantissas,  called  the  tabular  difference. 

In  all  cases,  the  mantissa  is  only  an  approximation. 
The  large  tables  o^  Adrien  Vlacq  give  the  logarithms  to 
ten  places  of  decimals  of.  all  numbers  from  1  to  100000. 
The  last  figure  is  given  within  one-half  a  unit  of  its  own 
order ;  that  is,  if  the  first  figure  of  the  part  not  given  is 
5  or  more,  then  the  last  figure  given  is  increased  by  1. 

TO  FIND  THE  LOGARITHM  OF  A  GIVEN  NUMBER. 

OlO.  If  the  significant  figures  of  the  number  are  the 
same  as  those  of  any  number  between  1000  and  11000, 
find  the  mantissa  in  the  table  and  prefix  the  proper  char- 
acteristic. 

For  example,  to  find  the  logarithm  of  1245,  find  124 
in  column  N;  in  the  same  line  and  in  column  5,  find 
5169 ;  prefix  .09  from  column  0 ;  then  prefix  the  charac- 
Triff.— 29. 


838  TRIGONOMETRY. 

teristic  3;  and  the  logarithm  of  1245  is  3.095169.  Sim- 
ilarly, 

log.  124500  =  5.095169, 

log.  .0001245  =  4.095169. 

If  the  significant  figures  are  those  of  a  number  less 
than  1000,  annex  ciphers  to  make  a  number  between 
1000  and  11000,  and  proceed  as  before.  For  example, 
the  logarithm  of  16  has  the  same  mantissa  as  the  log- 
arithm of  1600,  which  is  .204120.  Therefore,  the  log- 
arithm of  16  is.  1.204120. 

,  If  the  significant  figures  of  the  given  number  occupy 
more  places  than  the  numbers  in  the  table,  find  the 
mantissa  for  the  first  four  or  five  figures;  regard  the 
remaining  figures  as  a  decimal  fraction,  and  add  to  the 
mantissa  already  found  the  proportional  part  of  the  tab- 
ular difiierence. 

For  example,  to  find  the  logarithm  of  3.1416. 

The  mantissa  of  log.  3141  is     .     .     .    .497068, 
six-tenths  of  the  tabular  difference,  138,  is  83, 

the  characteristic  being  0, 497151  is  the 

logarithm  sought.  It  is  assumed  that  the  mantissa  of 
the  logarithm  of  3141.6  is  the  same  as  of  3141  increased 
by  six-tenths  of  the  difierence  between  the  mantissas  of 
3141  and  3142. 

To  find  the  logarithm  of  365.242. 

The  mantissa  of  log.  3652  is  =     562531, 

tab.  diff.  =  119 ;      119  X  .42  =  50. 

Therefore,      log.  365.242  =  2.562581. 

All  figures  beyond  the  six  places  of  decimals  are  re- 
jected from  the  calculations,  taking   care  that  the  last 


LOGAKIiHuS.  339 

figure  used  shall  be  the  nearest.  Thus,  six- tenths  of  138 
is  nearer  to  83  than  to  82. 

When  the  tabular  difference  varies  rapidly,  as  at  the 
beginning  of  the  table,  there  may  be  slight  errors  in  its 
use,  for  the  logarithms  do  not  vary  as  the  numbers.  On 
this  account,  for  all  numbers  between  10000  and  11000, 
it  is  better  to  use  the  last  two  pages  of  the  Table  instead 
of  the  first  ten  lines. 

If  the  given  •number  has  more  than  six  significant 
figures,  the  seventh  and  subsequent  figures  rarely  affect 
the  first  six  places  of  the  mantissa.  Thus,  the  logarithm 
of  365.24224  is,  to  six  places  of  decimals,  the  same  as 
the  logarithm  of  365.242. 

TO   FIND  THE  NUMBER,   ITS   LOGARITHM  BEING 
KNOWN. 

911.  If  the  mantissa  of  the  logarithm  is  the  same  as 
one  in  the  table,  take  the  corresponding  number,  and 
place  the  decimal  point  according  to  the  rule  of  the 
characteristic. 

If  the  given  mantissa  is  not  in  the  table,  find  that 
mantissa  in  the  table  which  is  next  less  than  the  given 
one,  and  take  the  corresponding  number.  Annex  to  this, 
two  figures  of  the  quotient  found  by  dividing  by  the  tab- 
ular difference,  the  excess  of  the  given  mantissa  over  the 
one  used.  Fix  the  decimal  point  by  the  rule  of  the 
characteristic. 

For  example,  to  find  the  number  whose  logarithm  is 
4.016234. 

The  next  less  mantissa  is  016197,  which  has  10380 
for  its  corresponding  number  (see  page  364).  The  dif- 
ference between  it  and  the  given  mantissa  is  37,  and  the 
tabular  difference  is  42. 


340  TRIGONOMETRY. 

Expressing  the  fraction  |J  decimally,  we  have  the  fig- 
ures 88  to  be  annexed  to  those  already  found,  making 
1038088,  the  significant  figures  of  the  required  number. 
The  characteristic  4  shows  that  the  first  significant  figure 
should  be  in  the  fifth  place.  Therefore,  10380.88  is 
the  number  sought. 

As  the  logarithms  are  only  approximations,  so  the 
number  found  can  only  be  said  to  be  true  for  six  or 
seven  places  of  figures.  When  a  grea'ter  degree  of  ex- 
actness is  required,  logarithms  must  be  used  of  more 
than  six  decimal  places.  These  may  be  calculated  by 
means  of  Table  II,  and  the  formula  given  with  it. 

MULTIPLICATION    AND    DIVISION. 

912.  Let  a;  and  y  represent  the  logarithms  of  M  and 
N  respectively. 

By  the  definition,     10"^  =  M. 

Similarly,  10^  =N. 

Multiplying  the  first  by  the  second, 

10^  +  ^=  MXN.    . 
Dividing  the  first  by  the  second, 

10^-^=M-f-N. 

That  is,  x-\-y  is  the  logarithm  of  the  product  of  M 
multiplied  by  N,  and  x  —  ^  is  the  logarithm  of  the  quo- 
tient of  M  divided  by  N.  Hence,  the  following  rules  for 
multipHcation  and  division  by  logarithms : 

To  multiply,  add  the  logarithms  of  the  factors.  The 
sum  is  the  logarithm  of  the  product. 


LOGARITHMS.  341 

To  divide^  subtract  the  logarithm  of  the  divisor  from 
that  of  the  dividend.  The  remainder  is  the  logarithm  of 
the  quotient. 

For  example,  to  find  the  product  of  2,  .000314,  and 
89.235. 

log.  2  =    .301030, 

log.  .000314  =  4.496930, 
log.  89.235    =  1.950535,     • 

The  sum,  2.748495  is  the  logarithm 
of  .0560396,  which  is  the  required  product,  true  to  six 
places  of  significant  figures. 

Again,  to  divide  2  by  .000314. 

log.  2  =    .301030, 

log.  .000314  =  4.496930, 
The  remainder,       3.804100  is  the  logarithm 
of  6369.43,  the  quotient,  true  to  six  places  of  figures. 

Care  must  be  exercised  in  the  additions  and  subtrac- 
tions, as  the  mantissas  are  all  positive  and  the  character- 
istics sometimes  negative. 

913.  It  saves  labor,  instead  of  subtracting  a  log- 
arithm, to  add  its  arithmetical  complement.  The  arith- 
metical complement  is  the  excess  of  10  over  the  loga- 
rithm. Let  I  represent  any  logarithm,  then  10  —  I  is 
its  complement.  If  10  —  I  is  added,  the  result  is  the 
same  as  when  I  is  subtracted  and  10  is  added.  There- 
fore, 

Each  time  that  an  arithmetical  complement  is  added, 
10  must  be  subtracted  from  the  result.  When  the  log- 
arithm is  itself  greater  than  10,  subtract  it  from  20  for 
the  complement,  and  add  20  to  the  result. 


342  TRIGONOMETRY. 

If  it  were  necessary  to  write  out  the  logarithm  in 
order  to  subtract  it  from  10,  there  would  be  little  saving 
of  labor,  but  the  complement  may  be  written  at  once, 
beginning  at  the  left,  and  subtracting  each  figure  of  the 
given  logarithm  from  9,  to  the  last  significant  figure 
which  is  to  be  subtracted  from  10.  This  method  is  par- 
ticularly useful  when  it  is  required  to  subtract  several 
logarithms. 

n    .    .        .        n  3456  X  89123 
For  example,  to  find  the  value  of  Q7f^Q  w    409-1  • 

log.  3456  =  3.538574, 

log.  89123  =  4.949990, 

a.  clog.  9753  =6.010862, 

a.  clog.  4321  =6.364416, 

log.  7.30873  =  .863842. 

The  sum  is  diminished  by  20,  for  the  complement 
twice  used.  Therefore,  7.30873  is  the  value  of  the  given 
fraction. 


INVOLUTION    AND    EVOLUTION. 
914.  Let  y  represent  the  logarithm  of  N.     Then, 

10^  =  N. 
Raising  both  members  to  the  x^^  power, 

10^^=N=^. 
Taking  the  x^^  root  of  both  members, 
10^=  v^N. 


LOGARITHMS.  343 

That  is,  xy  is  the  logarithm  of  the  x^^  power  of  N, 
and  I  is  the  logarithm  of  the  x^^  root  of  N.  Hence,  these 
rules  for  involution  and  evolution  by  logarithms : 

To  raise  a  number  to  a  required  power,  multiply  its 
logarithm  hy  the  exponent  of  the  power.  The  product  is 
the  logarithm  of  the  power. 

To  extract  any  root  of  a  number^  divide  its  logarithm 
by.  the  index  of  the  required  root.  The  quotient  is  the 
logarithm  of  the  root. 

In  making  this  division,  if  the  characteristic  of  the 
given  logarithm  is  negative,  and  is  not  exactly  divisible 
by  the  divisor,  then  increase  it  by  as  many  units  as  are 
needed  to  make  it  so  divisible,  prefixing  the  added  num- 
ber to  the  mantissa  as  an  integer.  The  result  is  not 
aflfected  by  thus  adding  the  same  number  to  both  the 
negative  and  positive  parts  of  the  logarithm. 

For  example,  to  find  the  fourth  root  of  J. 

log.  .5  =  1.698970. 

This  logarithm  is  equal  to  —4+3.698970,  in  which 
form  it  may  be  divided  by  4.  The  quotient  1.924742^  is 
the  logarithm  of  .840896,  which  is  the  fourth  root  of  J. 

015.  The  positive  or  negative  character  of  a  factor  is 
not  considered  in  the  use  of  logarithms.  The  proper 
sign  can  always  be  given  to  the  result,  according  to  the 
algebraic  principles. 

In  order  that  an  arithmetical  problem  may  be  solved 
by  logarithms,  it  should  not  contain  any  additions  or 
subtractions.  If,  for  example,  it  is  required  to  find  the 
sum  of  1^3  and  |/2,  each  root  may  be  found  separately 
by  the  aid  of  logarithms,  but  the  addition  must  be  made 
afterward  in  the  usual  manner. 


344  TRIGONOMETRY. 

Mathematicians  have  given  much  attention  to  the  con- 
struction of  such  trigonometrical  formulas  as  require  onlj 
the  operations  of  multiplication,  division,  involution,  and 
evolution.  For  examples  of  this,  see  Articles  866  and 
seq.  in  Plane  Triangles,  and  Articles  880  and  seq.  in 
Spherical  Triangles. 

EXERCISES. 
916. — 1.  Calculate  the  value  of  these  expressions: 
l/8932  X  .045726,     -|/7609 -r-^Io;     y^l3^  X  14«-f- 1.25«. 

2.  Find  the  area  of  a  circle,  the  radius  being  3  feet  (500). 

3.  What  is  the  diameter  of  a  circle  whose  circumference  is  314 
feet  3  inches  ? 

4.  What  is  the  area  of  a  triangle  whose  sides  are  417,  1493,  and 
1307  feet?    (390.) 

5.  The  diameter  of  the  earth  at  the  equator  being  41850000  feet, 
what  is  the  length  in  miles  of  one  degree  of  longitude  on  -the 
equator,  there  being  5280  feet  in  one  mile? 

6.  The  earth  being  a  sphere  with  a  radius  of  20890000  ft.,  how 
many  square  miles  are  there  in  its  surface? 

Additional  exercises  may  be  made  upon  the  formulas  of  Art.  807. 


TABLES 


LOGARITHMS  OF  NUMBERS, 

From  1  to  11000, 

LOGARITHMS  OF  168  PRIME  NUMBERS, 

To  15  PLACES  OF  Decimals, 

NATURAL  SINES  AND  TANGENTS, 

Fob  every  Ten  minutes, 

AND 

LOGARITHMIC  SINES  AND  TANGENTS, 

For  every  minute  of  the  quadrant. 


■  1 

Num.  100,  Log.  000. 

TABLE  I.— LOGARITHMS 

N. 
100 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

000000 

0434 

0868 

1301 

1734 

2166 

2598 

3029 

3461 

3891 

432 

101 

4321 

4751 

5181 

5609 

6038 

&466 

6894 

7321 

7748 

8174 

428 

102 

8600 

9026 

9451 

9876 

.0300 

.0724 

.1147 

.1570 

.1993 

.2415 

424 

103 

012837 

3259 

3680 

4100 

4521 

4940 

5m 

5779 

6197 

6616 

420 

104 

7033 

7451 

7868 

8284 

8700 

9116 

9532 

9947 

.0361 

.0775 

416 

105 

021189 

1603 

2016 

2428 

2841 

3252 

3664 

4075 

4486 

4896 

412 

106 

5306 

5715 

6125 

6533 

6942 

7350 

7757 

8164 

8571 

8978 

408 

107 

9384 

9789 

.0195 

.0600 

.1004 

.1408 

.1812 

.2210 

.2619 

.3021 

404 

108 

033421 

3826 

4227 

4628 

5029 

5430 

58,'M) 

6230 

6629 

7028 

401 

109 

7426 

7825 

8223 

8620 

9017 

9414 

9811 

.0207 

.0602 

.0998 

397 

110 

041393 

1787 

2182 

2.576 

2969 

3362 

3755 

4148 

4.540 

4932 

393 

111 

5323 

5714 

6105 

6495 

6885 

7275 

7664 

8053 

8442 

8830 

390 

112 

9218 

9606 

9993 

.0380 

.0766 

.1153 

.1538 

.1924 

.2309 

.2694 

386 

113 

053078 

3463 

3846 

42:30 

4613 

4996 

5378 

5760 

6142 

6524 

382 

114 

6905 

7286 

7666 

8046 

8426 

8805 

9185 

9563 

9942 

.0320 

379 

115 

060698 

1075 

1452 

1829 

2206 

2582 

2958 

3333 

3709 

4083 

376 

116 

4458 

4832 

5206 

5580 

5953 

6326 

6699 

7071 

7443 

7815 

373 

117 

8186 

8557 

8928 

9298 

9668 

.0038 

.0407 

.0776 

.1145 

.1514 

369 

118 

071882 

2250 

2617 

2985 

3352 

3718 

4085 

4451 

4816 

5182 

367 

119 

5547 

5912 

6276 

6640 

7004 

7368 

7731 

8094 

8457 

8819 

364 

120 

079181 

9543 

9904 

.0266 

.0626 

.0987 

.1347 

.1707 

.2067 

.2426 

360 

121 

082785 

3144 

3503 

3861 

4219 

4576 

4934 

5291 

5647 

6004 

358 

122 

6360 

6716 

7071 

7426 

7781 

8136 

8490 

8845 

9198 

9552 

356 

123 

9905 

.0258 

.0611 

.0963 

.1315 

.1667 

.2018 

.2370 

.2721 

.3071 

352 

124 

093422 

3772 

4122 

4471 

4820 

5169 

5518 

5866 

6215 

6562 

349 

125 

096910 

7257 

7604 

7951 

8298 

8644 

8990 

9335 

9681 

.0026 

346 

126 

100371 

0715 

1059 

1403 

1747 

2091 

2434 

2777 

3119 

3462 

344 

127 

3804 

4146 

4487 

4828 

5169 

5510 

5851 

6191 

6531 

6871 

341 

128 

7210 

7549 

7888 

8227 

8565 

8903 

9241 

9579 

9916 

.0253 

338 

129 

110590 

0926 

1263 

1599 

1934 

2270 

2605 

2940 

3275 

3609 

335 

130 

113943 

4277 

4611 

4944 

5278 

5611 

5943 

6276 

6608 

6940 

333 

131 

7271 

7603 

7934 

8265 

8595 

8926 

9256 

9586 

9915 

.0245 

330 

132 

120574 

0903 

1231 

1560 

1888 

2216 

2544 

2871 

3198 

3525 

328 

133 

3852 

4178 

4504 

4830 

5156 

5481 

5806 

6131 

6456 

6781 

325 

134 

7105 

7429 

T753 

8076 

8399 

8722 

9«5 

9368 

9690 

.0012 

323 

135 

130334 

0655 

0977 

1298 

1619 

1939 

2260 

2580 

2900 

3219 

321 

136 

3539 

3858 

4177 

4496 

4814 

5133 

5451 

5769 

6086 

6403 

318 

137 

6721 

7037 

7354 

7671 

7987 

a303 

8618 

8934 

9249 

9564 

310 

138 

9879 

.0194 

.0508 

.0822 

.1136 

.1450 

.1763 

.2076 

.2389 

.2702 

313 

139 

143015 

3327 

3639 

3951 

4263 

4574 

4885 

5196 

5507 

6818 

311 

140 

146128 

6438 

6748 

7058 

7367 

7676 

7986 

8294 

8603 

8911 

309 

141 

9219 

9527 

9835 

.0142 

.0449 

.0756 

.1063 

.1370 

.1676 

.1982 

307 

142 

152288 

2594 

2900 

3205 

3510 

3815 

4120 

4424 

4728 

,503'? 

305 

143 

5336 

5(>40 

5943 

6246 

6549 

6852 

7154 

7457 

7759 

8061 

303 

144 

8362 

8664 

8965 

9266 

9567 

9868 

.0168 

.0469 

.0769 

.1068 

301 

145 

161368 

1667 

1967 

2266 

2564 

2863 

3161 

3460 

3758 

4055 

299 

146 

4353 

4650 

4947 

5244 

5541 

5838 

6134 

6430 

6726 

7022 

297 

147 

7317 

7613 

7908 

S'Mi 

8497 

8792 

9086 

9380 

9674 

9968 

294 

148 

17Qg62 

0555 

0848 

1141 

14;^ 

1726 

2019 

2311 

2603 

2895 

293 

149 

3186 

3478 

3769 

4060 

4351 

4641 

4932 

5222 

5512 

5802 

291 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

346 


OF  NUMBERS. 

Num.  199,  Log.  300. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

150 

176091 

6381 

6670 

69.59 

7248 

7.536 

7825 

8113 

8401 

8689 

288 

151 

8977 

9264 

ft552 

9839 

.0126 

.0413 

.0699 

.0986 

.1272 

.1558 

287 

152 

181844 

2129 

2415 

2700 

2985 

3270 

3555 

3839 

4123 

4407 

285 

153 

4691 

4975 

52.59 

5542 

5825 

6108 

ft391 

6674 

6956 

72;^9 

283 

154 

7521 

7803 

8084 

8366 

8647 

8928 

9209 

9490 

9771 

.0051 

281 

155 

19a33^ 

0612 

0892 

1171 

1451 

1730 

2010 

2289 

2567 

2846 

279 

156 

31^ 

3403 

3681 

3959 

4237 

4514 

4792 

5069 

5346 

5623 

278 

157 

5900 

6176 

6453 

6729 

7005 

7281 

7556 

7832 

8107 

8382 

276 

1.58 

8657 

8932 

9206 

9481 

9755 

.0029 

.0303 

.0577 

.0850 

.1124 

274 

159 

201397 

1670 

1943 

2216 

2488 

2761 

3033 

3305 

3577 

3848 

272 

160 

204120 

4391 

4663 

4934 

5204 

5475 

5746 

6016 

6286 

6556 

270 

161 

6826 

7096 

7365 

7634 

7904 

8173 

8441 

8710 

8979 

9247 

269 

162 

9515 

9783 

.0051 

.0319 

.a586 

.0853 

.1121 

.1388 

.1654 

.1921 

267 

163 

212188 

2454 

2720 

2986 

32.52 

3518 

3783 

4049 

4314 

4579 

266 

164 

4844 

5109 

5373 

5638 

5902 

6166 

6430 

6694 

6957 

7221 

264 

165 

217484 

7747 

8010 

8273 

8536 

8798 

9060 

9323 

9585 

9816 

263 

166 

220108 

o;?70 

0631 

0892 

11.53 

1414 

1675 

1936 

2196 

2456 

261 

167 

2716 

2976 

32;^ 

3496 

3755 

4015 

4274 

4533 

4792 

5051 

259 

168 

5309 

5568 

5826 

6084 

6342 

6600 

6858 

7115 

7372 

7630 

258 

169 

7887 

8144 

8400 

8657 

8913 

9170 

9426 

9682 

9938 

.0193 

256 

170 

230449 

0704 

0960 

1215 

1470 

1724 

1979 

2234 

2488 

2742 

255 

171 

2996 

3250 

.3.504 

37.57 

4011 

4264 

4517 

4770 

5023 

5276 

253 

172 

5528 

5781 

60.« 

628.5 

6,537 

6789 

7041 

7292 

7514 

7795 

252 

173 

8046 

8297 

&548 

8799 

9049 

9299 

9550 

9800 

.0050 

.0300 

250 

174 

240549 

0799 

1048 

1297 

1546 

1795 

2044 

2293 

2541 

2790 

249 

175 

243038 

3286 

3534 

3782 

4a30 

4277 

4525 

4772 

5019 

5266 

248 

176 

5513 

5759 

6006 

62.52 

6499 

6745 

6991 

7237 

7482 

7728 

246 

177 

7973 

8219 

8464 

8709 

8954 

9198 

9443 

9687 

9932 

.0176 

245 

178 

250420 

0664 

0908 

1151 

1395 

1638 

1881 

2125 

2368 

2610 

243 

179 

2853 

3096 

3338 

3580 

3822 

4064 

4306 

4548 

4790 

5031 

242 

180 

255273 

5.514 

57,>5 

5996 

6237 

&477 

6718 

6958 

7198 

7439 

241 

181 

7679 

7918 

81.58 

8398 

8637 

8877 

9116 

9355 

9594 

9m 

239 

182 

260071 

0310 

0.548 

0787 

1025 

1263 

1501 

1739 

1976 

2214 

238 

im 

2451 

2688 

292;) 

3162 

3399 

3636 

3873 

4109 

4346 

4582 

237 

m 

4818 

5054 

5290 

5525 

5761 

5996 

6232 

6467 

6702 

6937 

235 

185 

267172 

7406 

7641 

7875 

8110 

8344 

8578 

8812 

9046 

9279 

234 

186 

9513 

9746 

9980 

.0213 

.0446 

.0679 

.0912 

.1144 

.1377 

.1609 

233 

187 

271842 

2074 

2306 

2.538 

2770 

3001 

3233 

3464 

3696 

3927 

232 

188 

4158 

4389 

4620 

48,50 

5081 

.5311 

5542 

5772 

6002 

6232 

2JJ0 

189 

6462 

6692 

6921 

7151 

7380 

7609 

7838 

8067 

8296 

8525 

229 

190 

278754 

8982 

9211 

9439 

9667 

9895 

.0123 

.0351 

.0578 

.0806 

228 

191 

281033 

1261 

1488 

1715 

1942 

2169 

2396 

2622 

2849 

3075 

227 

192 

3301 

3527 

37,53 

3979 

4205 

4431 

4656 

4882 

5107 

5332 

228 

193 

5557 

5782 

6007 

62;^2 

6456 

6681 

6905 

7130 

7354 

7578 

225 

194 

7802 

8026 

8249 

»473 

8696 

8920 

9143 

9366 

9589 

9812 

223 

195 

290035 

0257 

0480 

0702 

0925 

1147 

1369 

1591 

1813 

2034 

222 

196 

2256 

2478 

2699 

2920 

3141 

3363 

3584 

3804 

.  4025 

4246 

221 

197 

4466 

4687 

4907 

5127 

5347 

5567 

5787 

6007 

6226 

6446 

220 

198 

6665 

6884 

7104 

7323 

7542 

7761 

7979 

8198 

8416 

8635 

219 

199 

8853 

9071 

9289 

9507 

9725 

9943 

.0161 

.0378 

.0595 

.0813 

218 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8    9 

D. 

■  ■  ■  ■                ,           ..J 

347 


Num.  200,  Log.  301.   TABLE  T.— LOGARITHMS                 j 

N. 

0 

1 

2 

3 

4 

5 

6  1 

1 

7 

8 

9 

D. 

200 

30io;» 

1247 

1464 

1681 

1898 

2114 

2331 

2547 

2764 

2980 

217 

201 

3196 

3412 

3628 

3844 

4a59 

4275  j 

4491 

4706 

4921 

51,36 

216 

202i 

5aji 

5566 

5781 

5996 

6211 

6425 

&m 

6854 

7068 

7282 

215 

203 

7496 

7710 

7924 

8137 

83,51 

8564 

8778 

8991 

9204 

9417 

213 

204 

9630 

9843 

.00.56 

.0268 

.0481 

.0693 

.0906 

.1118 

.1330 

.1542 

212 

205 

3117&4 

1966 

2177 

2389 

1^600 

2812 

3023 

3284 

3445 

3656 

211 

206 

3867 

4078 

4289 

4499 

4710 

4920 

5l;30 

5340 

5551 

5760 

210 

207 

5970 

6180 

6390 

6599 

6809 

7018 

7227 

7436 

7646 

7854 

20f) 

208 

8063 

8272 

8481 

8689 

8898 

9106 

9314 

9522 

9730 

99;^ 

208 

209 

320146 

0354 

0562 

0769 

0977 

1184 

1391 

1598 

1805 

2012 

207 

210 

3t^l9 

2426 

2633 

2839 

3046 

3252 

3458 

3665 

3871 

4077 

206 

211 

4282 

4488 

4694 

4899 

5105 

5310 

5516 

5721 

5926 

6131 

2a5 

212 

6336 

6541 

6745 

6950 

7155 

7a59 

7563 

7767 

7972 

8176 

204 

213 

83801 

8583 

8787 

8991 

9194 

9398 

9601 

9805 

.0008 

.0211 

203 

214 

380414 

0617 

0819 

1022 

1225 

1427 

1630 

1832 

2034 

2236 

202 

215 

332438 

2640 

2842 

3044 

3246 

3447 

3649 

8850 

40,51 

4253 

202 

216 

4454 

46.55 

4856 

50.57 

5257 

5458 

5658 

5859 

6059 

6260 

201 

217 

6460 

6660 

6860 

7060 

7260 

7459 

7659 

7858 

8a58 

8257 

200 

218 

8456 

8656 

88.55 

9054 

9253 

9451 

96,50 

9849 

.0047 

.0246 

199 

219 

340444 

0642 

0841 

1039 

1237 

1435 

1632 

1830 

2028 

2225 

198 

220 

342423 

2620 

2817 

3014 

3212 

a409 

3606 

3802 

3999 

4196 

197 

221 

4392 

4589 

4785 

4981 

5178 

5374 

5570 

5766 

5962 

61,57 

196 

222 

6353 

6549 

6744 

6939 

7135 

7330 

7525 

7720 

7915 

8110 

195 

223 

8305 

8500 

8694 

8889 

9083 

9278 

9472 

9666 

9860 

.0054 

194 

224 

350248 

0442 

0636 

0829 

1023 

1216 

1410 

1603 

1796 

1989 

194 

225 

352183 

2375 

2568 

2761 

29.54 

3147 

aas9 

a582 

3724 

3916 

193 

226 

4108 

4301 

4493 

46&5 

4876 

5068 

5260 

5452 

5643 

5834 

192 

227 

6026 

6217 

6408 

6599 

6790 

6981 

7172 

7363 

7554 

7744 

191 

228 

79a5 

8125 

8316 

8506 

8696 

8886 

9076 

9266 

9456 

9646 

190 

229 

9835 

.0025 

.0215 

.0404 

.0593 

.0783 

.0972 

.1161 

.ia50 

.1539 

189 

230 

361728 

1917 

2105 

2294 

2482 

2671 

2859 

8048 

3236 

3424 

188 

231 

3612 

3800 

3988 

4176 

4363 

4551 

4739 

4926 

5113 

5301 

188 

232 

5488 

5675 

5862 

6049 

6236 

6423 

6610 

6796 

6983 

7169 

187 

233 

7356 

7542 

7729 

7915 

8101 

8287 

8473 

8659 

8845 

9030 

186 

234 

9216 

9401 

9587 

9772 

99.58 

.0143 

.0828 

.0513 

.0698 

.0883 

185 

235 

371068 

1253 

1437 

1622 

1806 

1991 

2175 

2360 

2544 

2728 

184 

236 

2912 

3096 

3280 

3464 

3647 

3831 

4015 

4198 

4382 

4565 

184 

%^1 

4748 

4932 

5115 

5298 

5481 

56ft4 

5846 

6029 

6212 

6394 

183 

238 

6577 

6759 

6942 

7124 

7306 

7488 

7670 

7852 

8034 

8216 

182 

239 

8398 

8580 

8761 

8943 

9124 

9306 

9487 

9668 

9849 

.0030 

181 

240 

380211 

0392 

0573 

0754 

0934 

1115 

1296 

1476 

1656 

1837 

181 

241 

2017 

2197 

23/V 

2557 

2737 

2917 

3097 

;3277 

a4,56 

36^36 

180 

242 

3815 

3995 

4174 

4,3,5;} 

4,533 

4712 

4891 

5070 

5249 

,5428 

179 

213 

5606 

5785 

5964 

6142 

ft^l 

6499 

6677 

6856 

7oai 

7212 

178 

244 

7390 

7568 

7746 

7923 

8101 

8279 

8456 

8634 

8811 

8989 

178 

245 

389166 

9343 

9.520 

9698 

9875 

.00,51 

.0228 

.0405 

.0582 

.0759 

177 

246 

3909a5 

1112 

1288 

1464 

1641 

1817 

1993 

2169 

2345 

2.521 

176 

247 

2697 

2873 

3048 

3224 

aioo 

a575 

3751 

3926 

4101 

4277 

176 

248 

4452 

4627 

4802 

4977 

5152 

5326 

5501 

5676 

5850 

6025 

175 

249 

6199 

6374 

6548 

6722 

6896 

7071 

7245 

7419 

7592 

7766 

174 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

348 


OF  NUMBERS.       Num.  299,  Log 

.476. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

250 

397940 

8114 

8287 

8461 

8634 

8808 

8981 

91.54 

9328 

9501 

173 

251 

9674 

9847 

.0020 

.0192 

.0365 

.a538 

.0711 

.0883 

.1056 

.1228 

173 

2.52 

401401 

1573 

1745 

1917 

2089 

2261 

24:33 

2t>05 

2777 

2949 

172 

253 

3121 

3292 

3464 

3G3.5 

3807 

3978 

4149 

4320 

4492 

46&S 

171 

251 

4834 

5005 

5176 

5346 

5,517 

5688 

5858 

6029 

6199 

6370 

171 

255 

406.540 

6710 

6881 

7a51 

7221 

7,391 

7561 

7731 

7901 

8070 

170 

256 

8240 

8410 

&579 

8749 

8918 

9087 

9257 

9126 

9.5a5 

9764 

169 

257 

9933 

.0102 

.0271 

.0440 

.0()09 

.0777 

.0946 

.1114 

.1283 

.1451 

169 

258 

411620 

1788 

19.56 

2124 

2293 

2461 

2629 

2796 

2964 

81,32 

168 

259 

3300 

3467 

3635 

3803 

3970 

4137 

4305 

4472 

4639 

4806 

167 

260 

414973 

5140 

5307 

5474 

5641 

5808 

5974 

6141 

6308 

6474 

167 

261 

6641 

6807 

6973 

7139 

7306 

7472 

7638 

7804 

7970 

8135 

166 

262 

8301 

8467 

8633 

8798 

8964 

9129 

9295 

9460 

9625 

9791 

165 

263 

99.56 

.0121 

.0286 

.0451 

.0616 

.0781 

.0^ 

.1110 

.1275 

.14:39 

165 

264 

421604 

1768 

1933 

2097 

2261 

2426 

2590 

2754 

2918 

3082 

1&4 

265 

423246 

3410 

a574 

3737 

3901 

4065 

4228 

4392 

4555 

4718 

1&4 

286 

4882 

5045 

5208 

5371 

om 

5697 

5860 

6023 

6186 

6349 

163 

267 

6511 

6674 

68;^ 

6999 

7161 

7321 

7486 

7648 

7811 

7973 

162 

268 

813.5 

8297 

8459 

8621 

8783 

8944 

9106 

9268 

9429 

9591 

162 

269 

9752 

9914 

.0075 

.0236 

.0.398 

.0559 

.0720 

.0881 

.1012 

.1203 

161 

270 

431364 

1525 

1685 

1846 

2007 

2167 

2328 

2488 

2649 

2809 

161 

271 

2969 

3130 

3290 

3450 

3610 

3770 

3930 

4090 

4249 

4409 

160 

272 

4569 

4729 

4888 

5048 

5207 

5,367 

5526 

5685 

5844 

6004 

159 

273 

6163 

6322 

6481 

6640 

6799 

69.57 

7116 

7275 

74.33 

7592 

159 

274 

7751 

7909 

8067 

8226 

8384 

8542 

8701 

8859 

9017 

9175 

158 

275 

439333 

9491 

9648 

9806 

9964 

.0122 

.0279 

.0437 

.0594 

.0752 

1.58 

276 

440909 

1066 

1224 

1381 

1.538 

1695 

1852 

2009 

2166 

2323 

1.57 

277 

2480 

2637 

2793 

29.50 

3106 

32<)3 

»119 

a576 

3732 

:3889 

1.57 

278 

4045 

4201 

4357 

4513 

4669 

4825 

4981 

5137 

.5293 

5149 

156 

279 

5604 

5760 

5915 

6071 

6226 

6382 

6,537 

6692 

6818 

7003 

155 

280 

447158 

7313 

7468 

7623 

7778 

7933 

8088 

8242 

8397 

S>52 

155 

281 

8706 

8861 

9015 

9170 

9,324 

9178 

9633 

9787 

9941 

.0095 

1-A 

282 

450249 

0403 

0557 

0711 

0865 

1018 

1172 

1326 

1479 

1633 

154 

283 

1786 

1940 

2093 

2247 

2400 

25,53 

2706 

2859 

3012 

3165 

153 

ay 

3318 

3471 

3624 

3777 

3930 

4082 

4235 

4387 

4540 

4692 

153 

2a5 

454845 

4997 

51.50 

5302 

5454 

5606 

5758 

5910 

6062 

6214 

152 

286 

6366 

6518 

6670 

6821 

6973 

7125 

7276 

7428 

7579 

7731 

152 

287 

7882 

8033 

8184 

83;56 

8487 

86:38 

8789 

8940 

9091 

9212 

151 

288 

9392 

9;^3 

9694 

9845 

999.5 

.0146 

.0296 

.0447 

.0,597 

.0748 

151 

289 

460898 

1018 

1198 

1348 

1499 

1649 

1799 

1948 

2098 

2218 

150 

290 

462398 

2548 

2697 

2847 

2997 

3146 

3296 

m5 

3594 

3744 

150 

291 

3893 

4042 

4191 

4;^0 

4490 

46:39 

4788 

4936 

5085 

5234 

149 

292 

5383 

5532 

5680 

5829 

5977 

6126 

6274 

ftl23 

6571 

6719 

149 

293 

6868 

7016 

7164 

7312 

7460 

760cS 

7756 

7904 

8052 

8200 

148 

m 

8^47 

849,5 

8643 

8790 

8938 

9085 

923:3 

9380 

9527 

9675 

148 

29.5 

469S22 

9969 

.0116 

.02ft3 

.0410 

.a5.57 

.0704 

.0851 

.0998 

.1145 

147 

290 

471292 

1488 

1.585 

1732 

1878 

2025 

2171 

2318 

2464 

2610 

146 

297 

27.56 

290:3 

3049 

3195 

3341 

3487 

3633 

3779 

3925 

4071 

146 

298 

4216 

4332 

4508 

4&53 

4799 

4944 

5090 

5235 

5381 

5526 

146 

299 

5671 

5816 

5902 

6107 

6252 

6397 

6542 

6687 

6832 

6976 

145 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

349 


Num.  300,  Log.  477. 

TABLE  I.— LOGARITHMS 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

300 

477121 

7266 

7411 

7555 

7700 

7Mi 

7989 

8133 

8278 

8422 

145 

301 

8566 

8711 

8855 

8999 

9143 

9287 

9431 

9575 

9719 

986:3 

144 

302 

480007 

0151 

0294 

0438 

0582 

072.5 

0869 

1012 

1156 

1299 

144 

303 

1443 

1586 

1729 

1872 

2016 

21.59 

2302 

2445 

2588 

2731 

143 

304 

2874 

3016 

3159 

3302 

3445 

a587 

3730 

3872 

4015 

4157 

143 

305 

484300 

4442 

45a5 

4727 

4869 

5011 

51.53 

5295 

5437 

5579 

142 

306 

5721 

5863 

6005 

6147 

6289 

6430 

a572 

6714 

68,55 

6997 

142 

307 

7138 

7280 

7421 

7563 

7704 

7845 

7986 

8127 

8269 

8410 

141 

308 

8551 

8692 

88.33 

8974 

9114 

9255 

9396 

9537 

9677 

9818 

141 

309 

9958 

.0099 

.0239 

.0380 

.0520 

.0661 

.0801 

.0941 

.1081 

.1222 

140 

310 

491362 

1502 

1642 

1782 

1922 

2062 

2201 

2341 

2481 

2621 

140 

311 

2760 

2900 

3040 

3179 

3319 

3458 

3.597 

3737 

3876 

4015 

139 

312 

41.5.5 

4294 

4433 

4.572 

4711 

48,50 

4989 

5128 

5267 

5406 

1,39 

313 

5544 

5683 

5822 

5960 

6099 

6238 

6376 

6515 

6653 

6791 

139 

314 

6930 

7068 

7206 

7344 

7483 

7621 

7759 

7897 

8035 

8173 

138 

315 

498311 

8448 

8586 

8724 

8862 

8999 

9137 

9275 

9412 

9550 

138 

316 

9687 

9824 

9962 

.0099 

.0236 

.0374 

.0511 

.0648 

.078,5 

.0922 

137 

317 

501059 

1196 

ims 

1470 

1607 

1744 

1880 

2017 

2154 

2291 

137 

318 

2427 

2.564 

2700 

2837 

2973 

3109 

3246 

3382 

3518 

3655 

136 

319 

3791 

3927 

4063 

4199 

43a5 

4471 

4607 

4743 

4878 

6014 

136 

320 

5051.50 

5286 

5421 

55.57 

5693 

5828 

5964 

6099 

6234 

6370 

136 

321 

6505 

6640 

6776 

6911 

7046 

7181 

7316 

7451 

7586 

7721 

135 

322 

7856 

7991 

8126 

8260 

8395 

8530 

8664 

8799 

89:34 

9068 

136 

323 

9203 

9337 

9471 

9606 

9740 

9874 

.0009 

.0143 

.0277 

.0411 

134 

324 

510545 

0679 

0813 

0947 

1081 

1215 

1349 

1482 

1616 

1750 

134 

32.5 

511883 

2017 

2151 

2284 

2418 

2551 

2684 

2818 

2951 

3084 

133 

320 

3218 

3351 

3484 

3617 

3750 

38m 

4016 

4149 

4282 

4415 

133 

327 

4.S48 

4681 

4813 

4946 

5079 

5211 

5344 

5476 

5609 

6741 

133 

328 

5874 

6006 

6139 

6271 

6403 

6535 

6668 

6800 

69,32 

7064 

132 

329 

7196 

7328 

7460 

7592 

7724 

7855 

7987 

8119 

8251 

8382 

132 

330 

518514 

8646 

8777 

8909 

9040 

9171 

9303 

9434 

9566 

9697 

131 

331 

9828 

9959 

.0090 

.0221 

.0353 

.0484 

.0615 

.0745 

.0876 

.1007 

131 

332 

5211.38 

1269 

1400 

1.530 

1661 

1792 

1922 

2053 

2183 

2314 

131 

m 

2444 

2575 

2705 

2835 

2966 

3096 

3226 

3356 

3486 

3616 

130 

334 

3746 

3876 

4006 

4136 

4266 

4396 

4526 

4656 

4785 

4915 

130 

335 

52.5045 

5174 

5304 

5434 

5563 

5693 

5822 

5951 

6081 

6210 

129 

336 

6a39 

6469 

6598 

6727 

6856 

698.5 

7114 

7243 

7372 

7501 

129 

337 

76;^ 

77,59 

7888 

8016 

8145 

8274 

8402 

85:31 

8660 

8788 

129 

338 

8917 

9045 

9174 

9302 

9430 

9559 

9687 

9815 

9943 

.0072 

128 

339 

530200 

0328 

0456 

0584 

0712 

0840 

0968 

1096 

1223 

1361 

128 

340 

531479 

1607 

1734 

1862 

1990 

2117 

2245 

2372 

2500 

2627 

128 

341 

2754 

2882 

3009 

3136 

3264 

3391 

3518 

3645 

3772 

3899 

127 

342 

4026 

4153 

4280 

4407 

45^ 

4661 

4787 

4914 

5041 

6167 

127 

343 

6294 

5421 

5547 

5674 

5800 

.5927 

605:3 

6180 

6306 

64:32 

126 

344 

6558 

6685 

6811 

69;^ 

7063 

7189 

7315 

7441 

7667 

7693 

126 

345 

537819 

7945 

8071 

8197 

a322 

8448 

8574 

8699 

8825 

8951 

126 

346 

9076 

9202 

9327 

9452 

9578 

9703 

9829 

9954 

.0079 

.0204 

125 

347 

540329 

04.55 

orm 

07a5 

om 

0955 

1080 

1205 

1330 

1454 

125 

348 

1579 

1704 

1829 

1953 

2078 

^W0.'{ 

2327 

24.52 

2576 

2701 

125 

349 

2825 

2950 

3074 

3199 

3323 

3447 

a571 

3696 

3820 

3944 

124 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

1 

350 


OF 

NUMBERS. 

Nnm.  399,  Log.  601. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

a50 

544068 

4192 

4316 

4440 

4564 

4688 

4812 

4936 

5060 

5183 

124 

351 

5307 

5431 

55.55 

5678 

5802 

5925 

6019 

6172 

6296 

6419 

124 

a52 

6543 

6666 

6789 

6913 

70.36 

7159 

7282 

7405 

7529 

7652 

123 

353 

7775 

7898 

8021 

8144 

8267 

8389 

8512 

8635 

8758 

8881 

123 

m 

9003 

9126 

9249 

9371 

9494 

9616 

9739 

9861 

9984 

.0106 

123 

355 

550228 

oa5i 

0473 

0595 

0717 

0840 

0962 

1084 

1206 

1328 

122 

356 

1450 

1572 

1694 

1816 

1938 

2060 

2181 

2303 

2425 

2547 

122 

a57 

2668 

2790 

2911 

30.33 

3155 

3276 

3398 

3519 

3640 

3762 

121 

a58 

3883 

4004 

4126 

4247 

4368 

4489 

4610 

4731 

4852 

4973 

121 

359 

5094 

5215 

5336 

54.57 

5578 

5699 

5820 

5940 

6061 

6182 

121 

360 

556303 

6423 

6.544 

6664 

6785 

69a5 

7026 

7146 

7267 

7387 

120 

361 

7507 

7627 

7748 

7868 

7988 

8108 

8228 

8349 

8469 

8589 

120 

362 

8709 

8829 

8948 

9068 

9188 

9308 

9428 

9548 

9667 

9787 

120 

363 

9907 

.0026 

.0146 

.0265 

.038.5 

.a5W 

.0624 

.0743 

.0863 

.0982 

119 

364 

561101 

1221 

1340 

1459 

1578 

1698 

1817 

1936 

2055 

2174 

119 

365 

FA9fm 

2412 

2.531 

2650 

2769 

2887 

3006 

3125 

8244 

8362 

119 

366 

3481 

3600 

3718 

3837 

39.55 

4074 

4192 

4311 

4429 

4548 

119 

367 

4666 

4784 

4903 

5021 

5139 

52.57 

5;37(i 

5494 

5612 

5730 

118 

368 

5848 

5966 

6084 

6202 

6320 

6437 

6555 

6673 

6791 

6909 

118 

369 

7026 

7144 

7262 

7379 

7497 

7614 

7732 

7849 

7967 

8084 

118 

370 

568202 

8319 

84.36 

8.5.54 

8671 

8788 

8905 

9023 

9140 

9257 

117 

371 

9374 

W91 

9608 

972.5 

9842 

99.59 

.0076 

.0193 

.0309 

.0426 

117 

372 

57a543 

0660 

0776 

0893 

1010 

1126 

1243 

ia59 

1476 

1592 

117 

378 

1709 

1825 

1942 

20.58 

2174 

2291 

2407 

2523 

2639 

2755 

116 

374 

2872 

2988 

3104 

3220 

3336 

^452 

3568 

3684 

3800 

3915 

116 

375 

574031 

4147 

4263 

4379 

4494 

4610 

4726 

4841 

4957 

5072 

116 

376 

5188 

5303 

5419 

5oM 

5650 

576,5 

5880 

5996 

6111 

6226 

115 

377 

6341 

64.57 

6.572 

6687 

6802 

6917 

7032 

7147 

7262 

7377 

115 

378 

7492 

7607 

7722 

7&« 

7951 

8066 

8181 

8295 

8410 

8525 

115 

379 

8639 

8754 

8868 

8983 

9097 

9212 

9326 

9441 

9555 

9669 

114 

380 

579784 

9898 

.0012 

.0126 

.0241 

.0355 

.0469 

.0583 

.0697 

.0811 

114 

381 

58092.5 

1039 

1153 

1267 

1381 

1495 

1608 

1722 

1836 

1950 

114 

382 

2063 

2177 

2291 

2404 

2518 

2631 

2745 

28o8 

2972 

3085 

114 

38:3 

3199 

asi2 

3426 

3539 

36.52 

3765 

3879 

3992 

4105 

4218 

113 

381 

4331 

4444 

4557 

4670 

4783 

4896 

5009 

5122 

5235 

5348 

113 

385 

5&5461 

5.574 

5B86 

5799 

5912 

6024 

61.37 

6250 

6362 

6475 

113 

386 

6587 

6700 

6812 

692.5 

70;37 

7149 

72(i2 

7374 

7486 

7599 

112  V 

387 

7711 

7823 

79.3,5 

8017 

8160 

8272 

8384 

8496 

8608 

8720 

112 

as8 

8832 

8944 

90;56 

9167 

9279 

9391 

9.J03 

9615 

9726 

9838 

112 

389 

9950 

.0061 

.0173 

.0284 

.0396 

.0507 

.0619 

.0730 

.0842 

.0953 

112 

390 

59106.5 

1176 

1287 

1399 

1510 

1621 

1732 

1843 

1955 

2066 

111 

391 

2177 

2288 

2399 

2510 

2621 

2732 

2843 

2954 

8064 

3175 

111 

392 

3286 

3397 

3508 

3618 

3729 

3840 

39.50 

4061 

4171 

4282 

111 

393 

ms 

4.503 

4614 

4724 

4834 

4945 

5055 

5165 

5276 

5386 

110 

394 

5496 

5606 

5717 

5827 

5937 

6047 

6157 

6267 

6377 

6487 

110 

395 

596597 

6707 

6817 

6927 

7037 

7146 

7256 

7366 

7476 

7586 

110 

396 

7695 

7805 

7914 

8024 

8134 

8243 

8353 

8462 

8572 

8681 

110 

397 

8791 

SJKX) 

9009 

9119 

9228 

9.337 

9446 

9556 

9665 

9774 

109 

398 

9883 

9992 

.0101 

.0210 

.0319 

.0428 

.0537 

.0646 

.0755 

.0864 

109 

399 

600973 

1082 

1191 

1299 

1408 

1517 

1625 

1734 

1843 

1951 

109 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

351 


- 

Num.  400,  Log.  602. 

TABLE  I.— LOGARITHMS 

N. 
400 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

602060 

2169 

2277 

2,386 

2494 

2603 

2711 

2819 

2928 

30:36 

108 

401 

3144 

32.53 

.3361 

3469 

3577 

3686 

3794 

3902 

4010 

4118 

108 

402 

4226 

4334 

4442 

45.50 

4658 

4766 

4874 

4982 

5089 

5197 

108 

403 

5305 

5413 

5521 

5628 

5736 

5844 

5951 

6059 

6166 

6274 

108 

404 

6381 

6489 

6.596 

6704 

6811 

6919 

7026 

7133 

7241 

7348 

107 

405 

607455 

7562 

7669 

7777 

7884 

7991 

8098 

8205 

a3i2 

8419 

107 

406 

8.526 

863:3 

8740 

8847 

8954 

9061 

9167 

9274 

9,381 

9488 

107 

407 

9594 

9701 

9808 

9914 

.0021 

.0128 

.02,34 

.0341 

.0447 

.0.554 

107 

408 

610660 

0767 

0873 

0979 

1086 

1192 

1298 

1405 

1511 

1617 

106 

409 

1723 

1829 

1936 

2042 

2148 

2254 

2360 

2466 

2572 

2678 

106 

410 

612784 

2890 

2996 

3102 

3207 

a313 

3419 

352,5 

3a30 

Sim 

106 

411 

3842 

3947 

4053 

4159 

4264 

4370 

4475 

4581 

4686 

4792 

106 

412 

4897 

5003 

5108 

5213 

5319 

5424 

5529 

5634 

5740 

5845 

105 

413 

5950 

6055 

6160 

6265 

6370 

6476 

6581 

6686 

6790 

6895 

lft5 

414 

7000 

7105 

7210 

7315 

7420 

7525 

7629 

7734 

7839 

7943 

105 

415 

618048 

8153 

8257 

8362 

8466 

8571 

8676 

8780 

8884 

8989 

105 

416 

9093 

9198 

9:302 

9406 

9511 

9615 

9719 

9824 

9928 

.0032 

104 

417 

620136 

0240 

0;344 

0448 

0.5,52 

0656 

0760 

0864 

09f)8 

1072 

104 

418 

1176 

1280 

1384 

1488 

1,592 

1695 

1799 

1903 

2007 

2110 

104 

419 

2214 

2318 

2421 

2525 

2628 

2732 

2835 

.2939 

3042 

3146 

104 

420 

623249 

3353 

34,56 

3559 

36ft3 

3766 

3869 

3973 

4076 

4179 

103 

421 

4282 

4385 

4488 

4591 

4695 

4798 

4901 

5004 

5107 

5210 

103 

422 

5312 

5415 

5,518 

5(521 

5724 

5827 

5929 

60:32 

6135 

6238 

103 

423 

6340 

6443 

6546 

6648 

6751 

6853 

6956 

7a58 

7161 

7263 

103 

424 

7366 

7468 

7571 

7673 

7775 

7878 

7980 

8082 

8185 

8287 

102 

425 

628389 

8491 

&593 

8695 

8797 

8900 

9002 

9104 

9206 

9308 

102 

426 

WIO 

9512 

9613 

9715 

9817 

9919 

.0021 

.0123 

.0224 

.0326 

102 

427 

630428 

0.5:30 

06:31 

07:33 

08:3.5 

09:36 

1088 

1139 

1241 

1342 

102 

428 

1444 

1.545 

1647 

1748 

1849 

1951 

2052 

21,5:3 

22,55 

2356 

101 

429 

2457 

2559 

2660 

2761 

2862 

2963 

3064 

3165 

3266 

3367 

101 

430 

6a3468 

3569 

3670 

3771 

3872 

3973 

4074 

4175 

4276 

4376 

101 

431 

4477 

4.578 

4679 

4779 

4880 

4981 

5081 

5182 

5283 

5383 

101 

432 

5484 

5,584 

508,5 

5785 

5886 

5986 

6087 

6187 

6287 

6388 

100 

433 

6488 

6588 

6688 

6789 

6889 

6989 

7089 

7189 

7290 

7390 

100 

434 

7490 

7590 

7690 

7790 

7890 

7990 

8090 

8190 

8290 

8389 

100 

435 

638489 

a589 

8689 

8789 

8888 

8988 

9088 

9188 

9287 

9387 

100 

436 

9486 

9586 

9686 

978.5 

9885 

9984 

.0084 

.0183 

.028:3 

.0382 

99 

437 

640481 

0581 

0680 

0779 

0879 

0978 

1077 

1177 

1276 

1375 

99 

4:i8 

1474 

1573 

1672 

1771 

1871 

1970 

2069 

2168 

2267 

2366 

99 

439 

2465 

2563 

2662 

2761 

2860 

2959 

3058 

3156 

3255 

33*4 

99 

440 

643453 

3551 

3650 

3749 

3847 

3946 

4044 

4143 

4242 

4340 

99 

441 

4439 

4.5:37 

4636 

4734 

4.S:32 

4931 

5029 

6127 

5226 

5.324 

98 

442 

&422 

5,521 

5619 

5717 

5815 

5913 

6011 

6110 

6208 

asoo 

98 

443 

6404 

6.502 

6600 

6698 

6796 

6894 

6992 

7089 

7187 

7285 

98 

444 

7383 

7481 

7579 

7676 

7774 

7872 

7969 

8067 

8165 

8262 

98 

445 

648360 

8458 

8555 

8653 

8750 

8848 

8945 

9043 

9140 

9237 

97 

446 

93a5 

9432 

9530 

9627 

9724 

9821 

9919 

.0016 

.0113 

.0210 

97 

447 

6,50308 

0405 

0502 

0599 

0696 

0793 

0890 

0987 

1084 

1181 

97 

448 

1278 

1375 

1472 

1569 

1666 

1762 

1859 

1956 

2053 

2150 

97 

449 

2246 

2343 

2440 

2536 

2633 

2730 

2826 

2923 

3019 

3116 

97 

N. 

0 

1 

2 

8 

4 

6 

6 

7 

8 

9 

D. 

352 


OF 

NUMBERS.       Num.  499,  Log 

.698. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

4;50 

653213 

3309 

3405 

3502 

3598 

3695 

3791 

3888 

3984 

4080 

96 

451 

4177 

4273 

4369 

446;5 

4.562 

4658 

4751 

48.50 

4M6 

5042 

96 

452 

5138 

5235 

5331 

5427 

5523 

5619 

5715 

5810 

5906 

6002 

96 

453 

6098 

6194 

6290 

6386 

6482 

6577 

6673 

6769 

6864 

6960 

96 

454 

7056 

7152 

7247 

7343 

7438 

7534 

7629 

7725 

7820 

7916 

96 

4r>5 

658011 

8107 

8202 

8298 

8393 

8488 

8584 

8679 

8774 

8870 

95 

456 

89&5 

9060 

91.55 

92.50 

9346 

9441 

9,536 

9631 

9726 

9821 

95 

457 

9916 

.0011 

.0106 

.0201 

.0296 

.0391 

.0486 

.0581 

.0676 

.0771 

95 

458 

66086.5 

0960 

ia5.5 

11.50 

1245 

1339 

14,34 

1.529 

1623 

1718 

95 

459 

1813 

1907 

2002 

2096 

2191 

2286 

2380 

2475 

2569 

2663 

95 

460 

662758 

2852 

2917 

3041 

31.35 

3230 

3324 

3418 

3512 

3607 

94 

461 

3701 

3795 

3889 

398;^ 

4078 

4172 

4266 

4360 

4454 

4548 

m 

462 

4642 

47.36 

48.30 

4924 

5018 

5112 

5206 

5299 

5393 

5487 

M 

463 

5581 

5675 

5769 

58()2 

5ft56 

6O50 

6143 

6237 

6:331 

6424 

M 

464 

6.518 

6612 

6705 

6799 

6892 

6986 

7079 

7173 

7266 

7360 

M 

4a5 

6674.53 

7.>46 

7640 

77.33 

7826 

7920 

8013 

8106 

8199 

82a3 

93 

466 

8386 

8479 

8ij72 

8665 

8759 

8852 

8iM5 

9038 

9131 

9224 

93 

467 

9317 

9410 

9,503 

9.596 

9689 

9782 

9875 

9967 

.0060 

.0153 

93 

468 

670246 

0339 

0431 

orm 

0617 

0710 

0802 

0895 

0988 

1080 

93 

469 

1173 

1265 

1358 

1451 

1543 

1636 

1728 

1821 

1913 

2005 

93 

470 

672098 

2190 

2283 

2;375 

2467 

2.560 

26,52 

2744 

2836 

2929 

92 

471 

3021 

3113 

320-5 

3297 

3390 

3482 

;3.574 

3666 

3758 

3850 

92 

472 

3942 

4034 

4126 

4218 

4310 

4402 

4494 

4.586 

4677 

4769 

92 

473 

4861 

4953 

5045 

5137 

5228 

5320 

&412 

5503 

5595 

5687 

92 

474 

5778 

5870 

5962 

6053 

6145 

6236 

6328 

fr419 

6511 

6602 

92 

475 

676694 

6785 

6876 

6968 

7059 

7151 

7242 

7333 

7424 

7516 

91 

476 

7607 

7698 

7789 

7881 

7972 

8063 

81M 

8245 

8336 

&427 

91 

477 

8.518 

8609 

8700 

8791 

8882 

8f)73 

9004 

9155 

9246 

9337 

91 

478 

9428 

9519 

9610 

9700 

9791 

9882 

9973 

.0063 

.0154 

.0245 

91 

479 

680.336 

0426 

0517 

0607 

0698 

0789 

0879 

0970 

1060 

1151 

91 

480 

681211 

ia32 

1422 

1513 

1603 

1693 

17*4 

1874 

1964 

2055 

90 

481 

2145 

223*5 

2;326 

2416 

2.506 

2596 

2686 

2777 

2867 

2957 

90 

482 

3047 

31;^ 

3227 

mn 

3407 

3497 

3587 

3677 

3767 

3857 

90 

48;^ 

3947 

4a37 

4127 

4217 

4:307 

4396 

4486 

4576 

4666 

4756 

90 

484 

4845 

4935 

5025 

5114 

5204 

5294 

5383 

5473 

5563 

5652 

90 

485 

685742 

5831 

5921 

6010 

6100 

6189 

6279 

6368 

6458 

6547 

89 

4m 

66.'i6 

6726 

6815 

6904 

6994 

7083 

7172 

7261 

7351 

7440 

89 

487 

7.529 

7618 

7707 

7796 

788(5 

7975 

8004 

8153 

8242 

8331 

89 

488 

8420 

8;309 

8.598 

8687 

8776 

8865 

89.53 

9042 

9131 

9220 

89 

489 

9309 

9398 

9486 

9575 

9664 

9753 

9841 

9930 

.0019 

.0107 

89 

490 

690196 

02&5 

0373 

0462 

0550 

0039 

0728 

0816 

0905 

0993 

89 

491 

1081 

1170 

12.58 

1347 

14:3.5 

1524 

1612 

1700 

1789 

1877 

88 

1 

492 

1965 

20.53 

2142 

22m 

2318 

2406 

2494 

2583 

2671 

2759 

88 

493 

2847 

29a5 

302:3 

3111 

3199 

3287 

3375 

3463 

3551 

3639 

88 

494 

3727 

3815 

3903 

3991 

4078 

4166 

4254 

4342 

4430 

4517 

88 

49.5 

694605 

4693 

4781 

4868 

4956 

5044 

5131 

5219 

5307 

5394 

88 

496 

5482 

5569 

5657 

5744 

5832 

5919 

6007 

6094 

6182 

6269 

87 

497 

6356 

&444 

6531 

6618 

6706 

6793 

6880 

6968 

7055 

7142 

87 

498 

7229 

7317 

7404 

7491 

7578 

7665 

7752 

7839 

7926 

8014 

87 

499 

8101 

8188 

8275 

8362 

8449 

8535 

8622 

8709 

8796 

8883 

87 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

Trig.— 30. 


353 


Num.  500,  Log.  698. 

TABLE  I.— LOGARITHMS 

1 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

500 

698970 

9057 

9144 

9231 

9317 

9404 

9491 

9578 

9664 

9751 

87 

501 

9838 

9924 

.0011 

.0098 

.0184 

.0271 

.oa58 

.0444 

.0531 

.0617 

87 

602 

700704 

0790 

0877 

0963 

1050 

1136 

1222 

1309 

1395 

1482 

86 

503 

1568 

1654 

1741 

1827 

1913 

1999 

2086 

2172 

2268 

2344 

86 

504 

2431 

2517 

2603 

2689 

2775 

2861 

2947 

3033 

3119 

3206 

86 

505 

703291 

3377 

3463 

3549 

3635 

3721 

3807 

3893 

3979 

4065 

86 

506 

4151 

4236 

4322 

4408 

4494 

4679 

4665 

4761 

4837 

4922 

86 

507 

5008 

5094 

5179 

5265 

5350 

5436 

5522 

5607 

5693 

5778 

86 

608 

5864 

5949 

6035 

6120 

6206 

6291 

6376 

6462 

6647 

6632 

85 

509 

6718 

6803 

6888 

6974 

7059 

7144 

7229 

7315 

7400 

7486 

85 

510 

707570 

7655 

7740 

7826 

7911 

7996 

8081 

8166 

8261 

8336 

86 

511 

8421 

8506 

8591 

8676 

8761 

8846 

8931 

9015 

9100 

918.5 

85 

512 

9270 

9355 

9440 

9524 

9609 

9694 

9779 

9863 

9948 

.0033 

85 

513 

710117 

0202 

0287 

0371 

0456 

0540 

0626 

0710 

0794 

0879 

85 

514 

0963 

1048 

1132 

1217 

1301 

1386 

1470 

1554 

1639 

1723 

84 

515 

711807 

1892 

1976 

2060 

2144 

2229 

2313 

2397 

2481 

2666 

84 

516 

2650 

2734 

2818 

2902 

2986 

3070 

3164 

3238 

3323 

3407 

84 

517 

3491 

3575 

3659 

3742 

3826 

3910 

3994 

4078 

4162 

4246 

84 

518 

4330 

4414 

4497 

4581 

4665 

4749 

4833 

4916 

5000 

5084 

M 

519 

5167 

5251 

5335 

5418 

5502 

6586 

5669 

5753 

5836 

5920 

84 

520 

716003 

6087 

6170 

62.54 

6337 

6421 

6504 

6588 

6671 

6754 

83 

521 

6838 

6921 

7004 

7088 

7171 

7254 

7338 

7421 

7604 

7,687 

83 

522 

7671 

7754 

7837 

7920 

8003 

8086 

8169 

8263 

8336 

8419 

83 

523 

8502 

8585 

8668 

8751 

8834 

8917 

9000 

9083 

9166 

9248 

83 

524 

9331 

9414 

9497 

9580 

9663 

9745 

9828 

9911 

9994 

.0077 

83 

525 

720159 

0242 

0325 

0407 

0490 

0673 

0656 

0738 

0821 

0903 

83 

526 

0986 

1068 

1151 

1233 

1316 

1398 

1481 

1563 

1646 

1728 

82 

527 

1811 

1893 

1975 

2058 

2140 

^?^^ 

2306 

2387 

2469 

2652 

82 

528 

2634 

2716 

2798 

2881 

2963 

3045 

3127 

3209 

3291 

3374 

82 

529 

3456 

3538 

3620 

3702 

3784 

3866 

3948 

4030 

4112 

4194 

82 

530 

724276 

4358 

4440 

4522 

4604 

4685 

4767 

4849 

4931 

5013 

82 

531 

5095 

5176 

5258 

5340 

5422 

5603 

5585 

5667 

5748 

5830 

82 

532 

5912 

5993 

6075 

6156 

6238 

6320 

6401 

6483 

6564 

6646 

82 

533 

6727 

6809 

6890 

6972 

7053 

7134 

7216 

7297 

7379 

7460 

81 

534 

7541 

7623 

7704 

7785 

7866 

7948 

8029 

8110 

8191 

8273 

81 

535 

728354 

8435 

8516 

8697 

8678 

8759 

8841 

8922 

9003 

9084 

81 

536 

9165 

9246 

9327 

9408 

9489 

9670 

9651 

9732 

9813 

9893 

81 

537 

9974 

.0055 

.0136 

.0217 

.0298 

.0378 

.0469 

.0540 

.0621 

.0702 

81 

538 

730782 

0863 

0944 

1024 

1105 

1186 

1266 

1347 

1428 

1608 

81 

539 

1589 

1669 

1750 

1830 

1911 

1991 

2072 

2152 

2233 

2313 

81 

540 

732394 

2474 

25;55 

2635 

2716 

2796 

2876 

2956 

3037 

3117 

80 

541 

3197 

3278 

3358 

3438 

3518 

3698 

3679 

3759 

38^^0 

3919 

80 

542 

3999 

4079 

4160 

4240 

4320 

4400 

4480 

4560 

4640 

4720 

80 

543 

4800 

4880 

4960 

5040 

5120 

5200 

5279 

5369 

5439 

5519 

80 

644 

5,599 

5679 

5759 

5838 

5918 

5998 

6078 

6157 

6237 

6317 

80 

645 

736397 

6476 

6556 

6635 

6715 

6795 

6874 

6954 

7034 

7113 

80 

546 

7193 

7272 

73*52 

7431 

7511 

7690 

7670 

7749 

7829 

7908 

79 

547 

7987 

8067 

8146 

8225 

8306 

8384 

8463 

8543 

8622 

8701 

79 

548 

8781 

8860 

8939 

9018 

9097 

9177 

9256 

9335 

9414 

9493 

79 

549 

9572 

9651 

9731 

9810 

9889 

9968 

.0047 

.0126 

.0205 

.0284 

79 

N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

D. 

354 


1 1 

OF 

NUMBERS. 

Nnm.  599,  Log.  778. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

5.50 

740363 

0442 

0521 

0600 

0678 

0757 

0836 

0915 

0994 

1073 

79 

551 

1152 

1230 

1.309 

1388 

1467 

1,546 

1624 

1703 

1782 

1860 

79 

552 

1939 

2018 

2096 

2175 

22,54 

2332 

2411 

2489 

2568 

2647 

79 

553 

2725 

2804 

2882 

2961 

3039 

3118 

3196 

3275 

3353 

3431 

78 

554 

3510 

3588 

3667 

3745 

3823 

3902 

3980 

4058 

4136 

4215 

78 

555 

744293 

4371 

4449 

4528 

4606 

4684 

4762 

4840 

4919 

4997 

78 

556 

5075 

51.53 

5231 

5309 

5387 

546.5 

5543 

5621 

5699 

5777 

78 

657 

5855 

5933 

6011 

6089 

6167 

6245 

6323 

6401 

6479 

6556 

78 

558 

6634 

6712 

6790 

,6868 

6945 

702;3 

7101 

7179 

7256 

7334 

78 

559 

7412 

7489 

7567 

7645 

7722 

7800 

7878 

7955 

8033 

8110 

78 

560 

748188 

8266 

&343 

8421 

8498 

8576 

8653 

8731 

8808 

8885 

77 

561 

8963 

9040 

9118 

9195 

9272 

9a50 

9427 

9504 

9582 

9659 

.  77 

562 

9736 

9814 

9891 

9968 

.004.5 

.0123 

.0200 

.0277 

.0364 

.0431 

77 

563 

7.50.508 

0586 

0663 

0740 

0817 

0894 

0971 

1048 

1125 

1202 

77 

564 

1279 

1356 

1433 

1510 

1.587 

1664 

1741 

1818 

1895 

1972 

77 

565 

752048 

2125 

2202 

2279 

2;i56 

24.33 

2.509 

2586 

2663 

2740 

77 

566 

2816 

2893 

2970 

3047 

312;] 

3200 

3277 

3353 

3430 

3506 

77 

567 

a583 

3660 

3736 

3813 

3889 

3966 

4042 

4119 

4195 

4272 

77 

568 

4348 

4425 

4.501 

4578 

46.54 

4730 

4807 

4883 

4960 

5036 

76 

569 

5112 

5189 

5265 

5341 

5417 

5494 

5570 

5646 

6722 

5799 

76 

570 

755875 

5951 

6027 

6103 

6180 

62,56 

6332 

ft408 

6484 

6560 

76 

571 

6636 

6712 

6788 

6864 

6940 

7016 

7092 

7168 

7244 

7320 

76 

572 

7396 

7472 

7.S48 

7624 

7700 

VVVo 

7851 

7927 

8003 

8079 

76 

573 

81.55 

sm 

8306 

8382 

sm 

&533 

8609 

8685 

8761 

8836 

76 

574 

8912 

8988 

9063 

9139 

9214 

9290 

9366 

9441 

9617 

9592 

76 

575 

759668 

9743 

9819 

9894 

9970 

.004.5 

.0121 

.0196 

.0272 

.0347 

75 

576 

760422 

0498 

0573 

0649 

0724 

0799 

0875 

0950 

1025 

1101 

75 

577 

1176 

12.51 

1.326 

1402 

1477 

1,5.52 

1627 

1702 

1778 

1853 

75 

578 

1928 

2003 

2078 

21.53 

2228 

2;«3 

2378 

2453 

2529 

2604 

76 

579 

2679 

2754 

2829 

2904 

2978 

3ft53 

3128 

3203 

3278 

3363 

75 

580 

763428 

3503 

a578 

3653 

3727 

3802 

3877 

3952 

4027 

4101 

75 

581 

4176 

42.51 

4326 

4400 

4475 

45^50 

4624 

4699 

4774 

4848 

75 

582 

4923 

4998 

5072 

5147 

5221 

5296 

5370 

6445 

5520 

5594 

75 

5m 

5669 

6743 

5818 

5892 

5966 

6041 

6115 

6190 

6264 

6338 

74 

584 

e413 

6487 

6562 

6636 

6710 

6785 

6859 

6933 

7007 

7082 

74 

585 

767156 

7230 

7.S04 

7379 

74.53 

7527 

7601 

7675 

7749 

7823 

74 

586 

7898 

7972 

8046 

8120 

8194 

8268 

8.342 

»416 

8490 

8564 

74 

587 

8638 

8712 

8786 

88(«) 

89.34 

9008 

9082 

91.56 

9230 

9303 

74 

588 

9377 

9451 

952.5 

9599 

9673 

9746 

9820 

9894 

9968 

.0042 

74 

589 

770115 

0189 

0263 

0336 

0410 

0484 

0557 

0631 

0705 

0778 

74 

590 

770852 

0926 

0999 

1073 

1146 

1220 

1293 

1367 

1440 

1614 

74 

591 

1587 

1G61 

1734 

1808 

1881 

19.55 

2028 

2102 

2175 

2248 

73 

592 

?^?9, 

2.395 

2468 

2542 

2615 

2688 

2762 

2835 

2908 

2981 

73 

593 

3a55 

3128 

3201 

3274 

3348 

3421 

3494 

3567 

3640 

3713 

73 

591 

3786 

3860 

3933 

4006 

4079 

4152 

4225 

4298 

4371 

4444 

73 

595 

774517 

4590 

4663 

4736 

4809 

4882 

49.55 

5028 

5100 

6173 

73 

596 

5246 

5319 

6392 

6465 

5538 

5610 

5683 

5756 

6829 

6902 

73 

597 

5974 

6047 

6120 

6193 

6265 

6338 

6411 

6483 

&566 

6629 

73 

598 

6701 

6774 

6846 

6919 

6992 

7064 

7137 

7209 

7282 

7354 

73 

599 

7427 

7499 

7572 

7644 

7717 

7789 

7862 

7934 

8006 

8079 

72 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

355 


Num 

600,  Log.  778. 

TABLE  I.— LOGARITHMS 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

778151 

8224 

8296 

8308 

8441 

8.513 

85.S.5 

8658 

8730 

8802 

72 

601 

8874 

8947 

9019 

9091 

9163 

9236 

9308 

9380 

9452 

9524 

72 

eo2 

£5)6 

9669 

9741 

9813 

9885 

99,57 

.0029 

.0101 

.0173 

.0245 

72 

608 

780317 

0389 

0461 

0533 

0605 

0677 

0749 

0821 

0893 

0965 

72 

604 

1037 

1109 

1181 

1253 

1324 

1396 

1468 

1540 

1612 

1681 

72 

60-) 

781755 

1827 

1899 

1971 

2042 

2114 

2186 

22.58 

2329 

2401 

72 

60i) 

2473 

2.544 

2616 

2688 

27,59 

2831 

2902 

2974 

3046 

3117 

72 

607 

31K9 

3260 

3:W2 

3403 

3475 

3546 

3618 

3689 

3761 

3832 

71 

608 

3904 

3975 

4046 

4118 

4189 

4261 

4332 

4403 

4475 

4546 

71 

609 

4617 

4689 

4760 

4831 

4902 

4974 

5045 

5116 

5187 

5259 

71 

610 

785330 

5401 

5472 

5543 

5615 

5686 

5757 

5828 

5899 

5970 

71 

611 

6041 

6112 

6183 

6254 

6325 

6396 

6467 

6,538 

6609 

6680 

71 

612 

6751 

6822 

6893 

6964 

70a5 

7106 

7177 

7248 

7319 

7390 

71 

613 

7460 

75:^1 

7602 

7673 

7744 

7815 

7885 

79,56 

8027 

8098 

71 

614 

8108 

8239 

8310 

8381 

8451 

8522 

8593 

8663 

8734 

8804 

71 

615 

788875 

8946 

9016 

9087 

9157 

9228 

9299 

9369 

9440 

9510 

71 

616 

9581 

9651 

9722 

9792 

9863 

9933 

.0004 

.0074 

.0144 

.0215 

70 

617 

790285 

0356 

0126 

0496 

0567 

0637 

0707 

0778 

0848 

0918 

70 

618 

0988 

1059 

1129 

1199 

1269 

1340 

1410 

1480 

1550 

1620 

70 

619 

1691 

1761 

1831 

1901 

1971 

2041 

2111 

2181 

2252 

2322 

70 

620 

792392 

2462 

2.532 

2602 

2672 

2742 

2812 

2882 

2952 

3022 

70 

621 

3092 

3162 

3231 

3301 

3371 

3441 

3511 

3581 

3651 

3721 

70 

622 

3790 

3860 

39,30 

4000 

4070 

4139 

4209 

4279 

4349 

4418 

70 

623 

4488 

4558 

4627 

4697 

4767 

4836 

4906 

4976 

5045 

5115 

70 

624 

5185 

5254 

5324 

5393 

5463 

5532 

5602 

5672 

5741 

5811 

70 

62;j 

795880 

5949 

6019 

6088 

6158 

6227 

6297 

6366 

6436 

6505 

69 

626 

6574 

6644 

6713 

6782 

68.52 

6921 

6990 

7060 

712',. 

7198 

69 

627 

7268 

7.337 

7406 

7475 

7.545 

7614 

7683 

7752 

7821 

7890 

69 

628 

7960 

8029 

8098 

8167 

82.36 

8305 

8374 

8443 

8513 

8582 

69 

629 

8651 

8720 

8789 

8858 

8927 

8996 

9065 

9134 

9203 

9272 

69 

630 

799341 

9409 

9478 

9,547 

9616 

9685 

9754 

9823 

9892 

9961 

69 

631 

800029 

0098 

0167 

0236 

0305 

0373 

0442 

0511 

0580 

0648 

69 

632 

0717 

0786 

0854 

0923 

0992 

1061 

1129 

1198 

1266 

133.5 

69 

633 

1404 

1472 

1,541 

1609 

1678 

1747 

1815 

1884 

1952 

2021 

69 

631 

2089 

2158 

2226 

2295 

2363 

2432 

2500 

2568 

2637 

2705 

69 

635 

802774 

2842 

2910 

2979 

3047 

3116 

3184 

3252 

3321 

3389 

68 

636 

3457 

3525 

a594 

3662 

3730 

3798 

3867 

39:35 

4003 

4071 

68 

637 

4139 

4208 

4276 

4344 

4412 

4480 

4,548 

4616 

4685 

47.53 

68 

638 

4821 

4889 

49.57 

5025 

5093 

5161 

5229 

5297 

5365 

54;33 

68 

639 

5501 

5569 

5637 

5705 

5773 

5841 

5908 

5976 

6044 

6112 

68 

640 

806180 

6248 

6316 

6384 

6451 

6,519 

&587 

6655 

6723 

6790 

68 

641 

68.58 

6926 

6994 

7061 

7129 

7197 

7264 

7.332 

7400 

7467 

68 

6i2 

7.5a5 

7603 

7670 

7738 

7806 

7873 

7941 

8008 

8076 

8143 

68 

643 

8211 

8279 

8;^46 

8414 

8481 

8,549 

8616 

8684 

8751 

8818 

67 

644 

8886 

8953 

9021 

9088 

91.56 

922;3 

9290 

9358 

9425 

9492 

67 

&15 

809560 

9627 

9694 

9762 

9829 

9896 

9964 

.0031 

.0098 

.0165 

67 

646 

8102;B 

o.%o 

0367 

0434 

0501 

0rj69 

0636 

0703 

0770 

0837 

67 

647 

0904 

0971 

1039 

1106 

1173 

1240 

1307 

1374 

1441 

1508 

67 

648 

1575 

1642 

1709 

1776 

1843 

1910 

1977 

2044 

2111 

2178 

67 

649 

2245 

2312 

2379 

2445 

2512 

2579 

2W6 

2713 

2780 

2847 

67 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

356 


OF  NUMBERS. 

Num.  699,  Log 

.845. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

3). 

650 

812913 

2980 

3047 

3114 

3181 

3247 

3314 

3381 

3448 

3514 

67 

651 

3581 

3648 

3714 

3781 

3848 

3914 

3981 

4048 

.  4114 

4181 

67 

652 

4248 

4314 

4381 

4447 

4514 

4581 

4647 

4714 

4780 

4847 

67 

653 

4913 

4980 

5046 

5113 

5179 

5246 

5312 

5378 

5445 

5511 

66 

654 

5578 

5614 

5711 

5777 

5843 

5910 

5976 

6042 

6109 

6175 

66 

655 

816241 

6308 

6374 

6440 

6506 

6573 

6639 

6705 

6771 

6838 

66 

656 

6904 

6970 

7036 

7102 

7169 

7235 

7301 

7367 

7433 

7499 

66 

657 

7565 

7631 

7698 

7764 

7830 

7896 

7962 

8028 

8094 

8160 

66 

658 

8226 

8292 

8358 

8424 

8490 

8556 

8622 

8688 

8754 

8820 

66 

659 

8885 

8951 

9017 

9083 

9149 

9215 

9281 

9346 

9412 

9478 

66 

660 

819.544 

9610 

9676 

9741 

9807 

9873 

9939 

.0004 

.0070 

.0136 

66 

661 

820201 

0267 

oa33 

0399 

0464 

0530 

0595 

0661 

0727 

0792 

66 

662 

0858 

0924 

0989 

1055 

1120 

1186 

1251 

1317 

1382 

1448 

66 

663 

1514 

1579 

1615 

1710 

1775 

1841 

1906 

1972 

2037 

2103 

65 

664 

2168 

2233 

2299 

2364 

2430 

2495 

2560 

?fiW 

2691 

2756 

65 

665 

822822 

2887 

2952 

3018 

3083 

3148 

3213 

3279 

3344 

ai09 

65 

666 

3474 

3539 

3605 

3670 

3735 

3800 

3865 

3980 

3996 

4061 

65 

667 

4126 

4191 

4256 

4321 

4386 

4451 

4516 

4581 

4646 

4711 

65 

668 

4776 

4841 

4906 

4971 

5036 

5101 

5166 

5231 

5296 

5361 

65 

669 

6426 

5491 

5556 

5621 

5686 

5751 

5815 

5880 

5945 

6010 

65 

670 

826075 

6140 

6204 

6269 

em 

6399 

0404 

6528 

6593 

66.58 

65 

671 

6723 

6787 

6852 

6917 

6981 

7046 

7111 

7175 

7240 

7305 

65 

672 

7369 

7434 

7499 

7563 

7628 

7692 

7757 

7821 

7886 

7951 

65 

673 

8015 

8080 

8144 

8209 

8273 

8338 

8402 

8467 

8531 

8595 

64 

674 

8660 

8724 

8789 

8853 

8918 

8982 

9046 

9111 

9175 

9239 

64 

675 

829304 

9368 

9432 

9497 

9561 

9625 

9690 

97,54 

9818 

9882 

64 

676 

9947 

.0011 

.0075 

.0139 

.0204 

.0208 

.0332 

.0396 

.0460 

.0525 

64 

677 

830589 

0653 

0717 

0781 

Ofy45 

0909 

0973 

1037 

1102 

1166 

64 

678 

1230 

1294 

1358 

1422 

1486 

1550 

1614 

1678 

1742 

1806 

64 

679 

1870 

1934 

1998 

2062 

2126 

2189 

2253 

2317 

2381 

^45 

64 

680 

832509 

2573 

2637 

2700 

2704 

2828 

2892 

2956 

3020 

3083 

64 

681 

3147 

3211 

3275 

aS38 

3402 

3466 

3530 

3593 

3657 

8721 

64 

682 

3784 

3848 

3912 

3975 

4039 

4103 

4166 

4230 

4294 

4357 

64 

683 

4421 

4484 

4548 

4611 

4675 

4739 

4802 

4866 

4929 

4993 

64 

684 

5056 

5120 

5183 

5^7 

5310 

5373 

5437 

5500 

5564 

5627 

63 

685 

835691 

5754 

5817 

5881 

6m 

6007 

6071 

6134 

6197 

6261 

63 

686 

6324 

6387 

6451 

6514 

6577 

6041 

6704 

6767 

6830 

6894 

63 

687 

6957 

7020 

7083 

7146 

7210 

7273 

7336 

7399 

7462 

7525 

63 

688 

7588 

7652 

7715 

7778 

7841 

7904 

7967 

8030 

8093 

8156 

63 

689 

8219 

8282 

8345 

8408 

8471 

8534 

8597 

8660 

8723 

8786 

63 

690 

838&49 

8912 

8975 

9038 

9101 

9164 

9227 

9289 

9352 

9415 

63 

691 

9478 

9541 

9604 

9667 

9729 

9792 

9855 

9918 

9981 

.0043 

63 

692 

840106 

0169 

02132 

0294 

0357 

0420 

0482 

0545 

0608 

0671 

63 

693 

0733 

0796 

0&39 

0921 

0984 

1046 

1109 

1172 

1234 

1297 

63 

694 

1359 

1422 

1485 

1547 

1610 

1672 

17aj 

1797 

1860 

1922 

63 

695 

-841985 

2047 

2110 

2172 

2235 

2297 

2360 

2422 

2484 

2547 

62 

690 

2609 

2672 

2734 

2796 

2859 

2921 

2983 

3046 

3108 

3170 

62 

697 

3233 

3295 

3357 

3420 

3482 

3544 

3606 

3669 

3731 

3793 

62 

698 

3855 

3918 

3980 

4042 

4104 

4166 

4229 

4291 

4a53 

4415 

62 

699 

4477 

4X39 

4601 

4664 

4726 

4788 

4850 

4912 

4974 

5036 

62 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

357 


^ 

Num.  700,  Log.  845. 

TABLE  I.— LOGARITHMS 

N. 
700 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

845098 

5160 

5222 

5284 

5346 

5-108 

5470 

5532 

5594 

5656 

62 

701 

5718 

5780 

5842 

5904 

5966 

6028 

6090 

6151 

6213 

6275 

62 

702 

6337 

6399 

6461 

6523 

6585 

6646 

6708 

6770 

6832 

6894 

62 

703 

6955 

7017 

7079 

7141 

7202 

7264 

7326 

7388 

7449 

7511 

62 

7W 

7573 

7634 

7696 

7758 

7819 

7881 

7943 

8004 

8066 

8128 

62 

705 

848189 

8251 

8312 

8374 

8435 

8497 

8559 

8620 

8682 

8743 

62 

706 

8805 

8866 

8928 

8989 

9051 

9112 

9174 

9235 

9297 

9358 

61 

707 

9419 

9481 

9542 

9604 

9665 

9726 

9788 

9849 

9911 

9972 

61 

708 

850033 

0095 

0156 

0217 

0279 

0340 

0401 

0462 

0524 

0585 

61 

709 

0646 

0707 

0769 

0830 

0891 

0952 

1014 

1075 

1136 

1197 

61 

710 

8512-58 

1320 

1381 

1442 

1503 

1.564 

1625 

1686 

1747 

1809 

61 

711 

1870 

1931 

1992 

2053 

2114 

2175 

2236 

2297 

2-358 

2419 

61 

712 

2480 

2541 

2602 

2663 

2724 

2785 

2846 

2907 

2968 

3029 

61 

713 

3090 

3150 

3211 

3272 

3333 

3394 

34.55 

3516 

3577 

3637 

61 

714 

3698 

3759 

3820 

3881 

3941 

4002 

4063 

4124 

4185 

4245 

61 

715 

854306 

4367 

4428 

4488 

4549 

4610 

4670 

4731 

4792 

4852 

61 

716 

4913 

4974 

5034 

5095 

5156 

5216 

5277 

5337 

5398 

5459 

61 

717 

5519 

5580 

5640 

5701 

5761 

5822 

5882 

5943 

6003 

6064 

61 

718 

6124 

6185 

6245 

6306 

6366 

6427 

6487 

6548 

6608 

6668 

60 

719 

6729 

6789 

6850 

6910 

6970 

7031 

7091 

7152 

7212 

7272 

60 

720 

857332 

7393 

7453 

7513 

7574 

7634 

7694 

7755 

7815 

7875 

60 

721 

7935 

7995 

80.56 

8116 

8176 

8236 

8297 

8,357 

8417 

8477 

60 

722 

8537 

8597 

8657 

8718 

8778 

8838 

8898 

8958 

9018 

9078 

60 

723 

9138 

9198 

9258 

9318 

9379 

9439 

9499 

9559 

9619 

9679 

60 

724 

9739 

9799 

9859 

9918 

9978 

.0038 

.0098 

.0158 

.0218 

.0278 

60 

725 

860338 

0398 

0458 

0518 

0578 

0637 

0697 

0757 

0817 

0877 

60 

726 

0937 

0996 

1056 

1116 

1176 

1236 

1295 

1355 

1415 

1475 

60 

727 

1534 

1594 

1654 

1714 

1773 

183:3 

1893 

1952 

2012 

2072 

60 

728 

2131 

2191 

2251 

2310 

2370 

2430 

2489 

2-549 

2608 

^668 

60 

729 

2728 

2787 

2847 

2906 

2966 

3025 

3085 

3144 

3204 

3263 

60 

730 

86a323 

3382 

3442 

3501 

3561 

3620 

3680 

3739 

3799 

3858 

59 

731 

3917 

3977 

4036 

4096 

4155 

4214 

4274 

4333 

4392 

4452 

59 

732 

4511 

4570 

4630 

4689 

4748 

4808 

4867 

4926 

4985 

5045 

59 

733 

5104 

5163 

5222 

5282 

5341 

5400 

5459 

5519 

5578 

5637 

59 

734 

5696 

5755 

5814 

5874 

5933 

5992 

6051 

6110 

6169 

6228 

59 

735 

866287 

6346 

6405 

6465 

6524 

6583 

6642 

6701 

6760 

6819 

59 

786 

6878 

6937 

6996 

7055 

7114 

7173 

7232 

7291 

7350 

7409 

59 

737 

7467 

7526 

7585 

7644 

7703 

7762 

7821 

7880 

7939 

7998 

59 

738 

8056 

8115 

8174 

8233 

8292 

8350 

8409 

ms 

8527 

8586 

59 

739 

8644 

8703 

8762 

8821 

8879 

8938 

8997 

9056 

9114 

9173 

59 

740 

869232 

9290 

9349 

9408 

9466 

9525 

9584 

9642 

9701 

9760 

59 

741 

9818 

9877 

9935 

9994 

.0053 

.0111 

.0170 

.0228 

.0287 

.0345 

59 

742 

870404 

0462 

0521 

0579 

0638 

0696 

0755 

0813 

0872 

0930 

58 

743 

0989 

1047 

1106 

1164 

1223 

1281 

1339 

1398 

1456 

1515 

58 

744 

1573 

1631 

1690 

1748 

1806 

1865 

1923 

1981 

2040 

2098 

58 

745 

872156 

2215 

2273 

2331 

2389 

2448 

2506 

2564 

2622 

2681 

58 

74€ 

2739 

2797 

2855 

2913 

2972 

30:30 

3088 

3146 

3204 

3262 

58 

747 

3321 

a379 

3437 

3495 

3.353 

3611 

3669 

3727 

3785 

3844 

58 

74^ 

3902 

3960 

4018 

4076 

4134 

4192 

4250 

4308 

4366 

4424 

58 

74i 

4482 

4540 

4598 

4656 

4714 

•  4772 

4830 

4888 

4945 

5003 

58 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

358 


OF 

NUMBERS. 

Num.  799,  Log.  903. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

750 

875061 

5119 

5177 

5235 

5293 

5351 

5409 

5466 

6524 

5682 

58 

751 

5640 

5698 

5756 

5813 

5871 

.5929 

5987 

6045 

6102 

6160 

58 

752 

6218 

6276 

6333 

6391 

6449 

6507 

6564 

6622 

6680 

6737 

58 

7.53 

6795 

6853 

6910 

6968 

7026 

7083 

7141 

7199 

7256 

7314 

58 

754 

7371 

7429 

7487 

7544 

7602 

7659 

7717 

7774 

7832 

7889 

58 

755 

877947 

8004 

8062 

8119 

8177 

8234 

8292 

8349 

8407 

8464 

57 

756 

8522 

8579 

8637 

8694 

8752 

8809 

8866 

8924 

8981 

9039 

57 

757 

9096 

9153 

9211 

9268 

9325 

9383 

9440 

9497 

9555 

9612 

67 

758 

9669 

9726 

9784 

9841 

9898 

9956 

.0013 

.0070 

.0127 

.0185 

57 

759 

880242 

0299 

0356 

0413 

0471 

0528 

0585 

0642 

0699 

0756 

57 

760 

880814 

0871 

0928 

09&5 

1042 

1099 

1156 

1213 

1271 

1328 

57 

761 

138.5 

1442 

1499 

1.5.56 

1613 

1670 

1727 

IIM 

1841 

1898 

57 

762 

1955 

2012 

2069 

2126 

2183 

2240 

2297 

2354 

2411 

2468 

67 

763 

2525 

2.581 

2638 

2695 

2752 

2809 

2866 

2923 

2980 

3037 

57 

764 

3093 

3150 

3207 

3264 

3321 

3377 

34S4 

»491 

3548 

3605 

57 

765 

883661 

3718 

3775 

3832 

8888 

3^5 

4002 

4059 

4115 

4172 

57 

766 

4229 

4285 

4342 

4399 

4455 

4512 

4.569 

4625 

4682 

4739 

57 

767 

47a5 

4852 

4909 

4965 

5022 

5078 

5135 

5192 

5248 

5305 

57 

768 

5361 

5418 

5474 

5531 

5587 

5644 

5700 

5757 

5813 

5870 

57 

769 

5926 

5983 

6039 

6096 

6152 

6209 

6265 

6321 

6378 

6434 

56 

770 

886491 

6547 

6604 

6660 

6716 

6773 

6829 

6885 

6942 

6998 

56 

771 

7054 

7111 

7167 

7223 

7280 

7.336 

7392 

7449 

7505 

7561 

56 

772 

7617 

7674 

7730 

7786 

7812 

7898 

7955 

8011 

8067 

8123 

56 

773 

8179 

8236 

8292 

8348 

8404 

8460 

8516 

8573 

8629 

8685 

66 

774 

8741 

8797 

8853 

8909 

8f)65 

9021 

yovv 

9ia4 

9190 

9246 

56 

775 

889302 

9358 

9414 

9470 

9.526 

9582 

9638 

9694 

9760 

9806 

56 

776 

9862 

9918 

9974 

.oo;^j 

.0086 

.0141 

.0197 

.0253 

.0309 

.ft365 

56 

777 

890421 

0477 

0533 

0.589 

0645 

0700 

0756 

0812 

0868 

0924 

56 

778 

0980 

1035 

1091 

1147 

1203 

12.59 

1314 

i;^o 

1426 

1482 

66 

779 

1537 

1593 

1649 

1705 

1760 

1816 

1872 

1928 

1983 

2039 

56 

780 

892095 

2150 

2206 

2262 

2317 

2373 

2429 

2484 

2540 

2595 

56 

781 

2651 

2707 

2762 

2818 

2873 

2929 

2985 

3040 

3096 

3151 

66 

782 

3207 

3262 

3318 

3373 

3429 

3484 

3540 

3595 

3651 

3706 

56 

783 

3762 

3817 

3873 

3928 

3984 

4039 

4094 

4150 

4205 

4261 

55 

784 

4316 

4371 

4427 

4482 

4538 

4593 

4648 

4704 

4759 

4814 

65 

785 

894870 

492.5 

4980 

5036 

5091 

5146 

5201 

62,57 

5312 

6367 

55 

786 

5423 

5478 

5533 

5588 

6644 

5699 

6754 

6809 

68&4 

6920 

55 

787 

5975 

6030 

6085 

6140 

6195 

6251 

6306 

6361 

6416 

6471 

55 

788 

6526 

6581 

6636 

6692 

6747 

6802 

6857 

6912 

6967 

7022 

55 

789 

7077 

7132 

7187 

7242 

72^ 

7352 

7407 

7462 

7517 

7572 

65 

790 

897627 

7682 

7737 

7792 

7847 

7902 

7957 

8012 

8067 

8122 

55 

791 

8176 

8231 

8286 

8^41 

8396 

8451 

8.506 

8561 

8615 

8670 

65 

792 

8725 

8780 

8835 

8890 

8944 

8999 

9054 

9109 

9164 

9218 

55 

793 

9273 

9328 

9383 

9437 

9492 

9547 

9602 

9656 

9711 

9766 

55 

794 

9821 

9875 

9930 

9985 

.0039 

.0094 

.0149 

.0203 

.0258 

.0312 

65 

795 

900367 

0422 

0476 

0531 

0586 

0640 

0695 

0749 

0804 

0859 

55 

796 

0913 

0968 

1022 

1077 

1131 

1186 

1240 

1295 

1349 

1404 

65 

797 

1458 

1513 

1567 

1622 

1676 

1731 

1785 

1840 

1894 

1948 

54 

798 

2003 

2057 

2112 

2166 

2221 

2275 

2329 

2384 

2438 

2492 

54 

799 

2547 

2601 

2655 

2710 

2764 

2818 

2873 

2927 

2981 

3036 

54 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

359 


Num.  800,  Log.  903. 

TABLE  L— LOGARITHMS 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

800 

903090 

3144 

3199 

3253 

3307 

3361 

3416 

3470 

3524 

3578 

54 

801 

3633 

3687 

3741 

3795 

3849 

3904 

3958 

4012 

4066 

4120 

54 

802 

4174 

4229 

4283 

4337 

4391 

4445 

4499 

4553 

4607 

4661 

54 

803 

4716 

4770 

4824 

4878 

4932 

4986 

5040 

5094 

5148 

5202 

54 

804 

5256 

5310 

5364 

5418 

5472 

5526 

5580 

5634 

5688 

5742 

54 

805 

905796 

5850 

5904 

5958 

6012 

6066 

6119 

6173 

6227 

6281 

54 

806 

6335 

6389 

6443 

6497 

6551 

6604 

66.58 

6712 

6766 

6820 

54 

807 

6874 

6927 

6981 

7035 

7089 

7143 

7196 

7250 

7304 

7358 

54 

808 

7411 

7465 

7519 

7573 

7626 

7680 

7734 

7787 

7841 

7895 

54 

809 

7949 

8002 

8056 

8110 

8163 

8217 

8270 

8324 

8378 

8431 

54 

810 

908485 

8539 

8592 

8646 

8699 

8753 

8807 

8860 

8914 

8967 

54 

811 

9021 

9074 

9128 

9181 

9235 

9289 

9342 

9;396 

9449 

9503 

54 

812 

9556 

9610 

9663 

9716 

9770 

9823 

9877 

9930 

9984 

.0037 

53 

813 

910091 

0144 

0197 

0251 

0304 

oa58 

0411 

0464 

0518 

0571 

53 

814 

0624 

0678 

0731 

0784 

0838 

0891 

0944 

0998 

1051 

1104 

53 

815 

911158 

1211 

1264 

1317 

1371 

1421 

1477 

1.530 

1584 

1637 

53 

816 

1690 

1743 

1797 

1850 

1903 

1956 

2009 

2063 

2116 

2169 

53 

817 

2222 

2275 

2328 

2381 

2435 

2188 

2541 

2594 

2647 

2700 

53 

818 

2753 

2806 

2859 

2913 

2966 

3019 

3072 

3125 

3178 

3231 

53 

819 

3284 

3337 

3390 

3443 

3496 

3549 

3602 

3655 

3708 

3761 

53 

820 

913814 

3867 

3920 

3973 

4026 

4079 

4132 

4184 

4237 

4290 

53 

821 

4343 

4396 

4449 

4502 

4.555 

4608 

4660 

4713 

4766 

4819 

53 

822 

4872 

4925 

4977 

5030 

5083 

6136 

5189 

5241 

5294 

5347 

53 

823 

5400 

5453 

5505 

5558 

5611 

5664 

5716 

5769 

5822 

5875 

53 

824 

5927 

5980 

6033 

6085 

6138 

6191 

6243 

6296 

6349 

6401 

53 

825 

916454 

6507 

6559 

6612 

6664 

6717 

6770 

6822 

6875 

6927 

53 

826 

6980 

7033 

7085 

7138 

7190 

7243 

7295 

7348 

7400 

7453 

53 

827 

7506 

7558 

7611 

7663 

7716 

7768 

7820 

7873 

7925 

7978 

52 

828 

8030 

8083 

8135 

8188 

8240 

8293 

8345 

8397 

8450 

8502 

52 

829 

8555 

8607 

8659 

8712 

8764 

8816 

8869 

8921 

8973 

9026 

52 

830 

919078 

9130 

9183 

9235 

9287 

9340 

9892 

9444 

9496 

9549 

52 

831 

9601 

9653 

9706 

9758 

9810 

9862 

9914 

9967 

.0019 

.0071 

52 

832 

920123 

0176 

0228 

0280 

0332 

0384 

0436 

0489 

0541 

0593 

52 

833 

0645 

0697 

0749 

0801 

0853 

0906 

0958 

1010 

1062 

1114 

52 

834 

1166 

1218 

1270 

1322 

1374 

1426 

1478 

1530 

1582 

1634 

52 

835 

921686 

1738 

1790 

1842 

1894 

1946 

1998 

2050 

2102 

2154 

52 

836 

2206 

2258 

2310 

2362 

2414 

2466 

2518 

2570 

2622 

2674 

52 

837 

2725 

2777 

2829 

2881 

2933 

2985 

3037 

3089 

3140 

3192 

52 

838 

3244 

3296 

3348 

3399 

S451 

3503 

&555 

3607 

3658 

3710 

52 

839 

3762 

3814 

3865 

3917 

3969 

4021 

4072 

4124 

4176 

4228 

52 

840 

924279 

4331 

4383 

4434 

4486 

4.538 

4589 

4641 

4693 

4744 

52 

841 

4796 

4848 

4899 

4951 

5003 

5054 

5106 

5157 

5209 

5261 

52 

842 

5312 

5364 

5415 

5467 

5518 

5570 

5621 

5673 

5725 

5776 

52 

843 

5828 

5879 

5931 

5982 

6034 

6085 

6137 

6188 

6240 

6291 

51 

844 

6342 

6394 

6445 

6497 

6518 

6600 

6651 

6702 

6754 

6805 

51 

845 

926857 

6908 

6959 

7011 

7062 

7114 

7165 

7216 

7268 

7319 

51 

846 

7370 

7422 

7473 

7524 

7576 

7627 

7678 

7730 

7781 

7832 

51 

847 

7883 

7935 

7986 

8037 

8088 

8140 

8191 

8242 

8293 

8345 

51 

848 

8396 

8447 

8498 

8549 

8601 

8652 

8703 

8754 

8805 

8857 

51 

849 

8908 

8959 

9010 

9061 

9112 

9163 

9215 

9266 

9317 

9368 

51 

1 , 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

360 


1                                                      ' 

OF 

NUMBERS. 

Nxun.  899,  Log 

.954. 

N. 
850 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

9879 

D. 

61 

929419 

9470 

9521 

9572 

9623 

9674 

972,5 

9776 

9827 

8.51 

9930 

9981 

.0032 

.0083 

.01,34 

.0186 

.0236 

.0287 

.0338 

.0.389 

51 

852 

930440 

0491 

0542 

0592 

0643 

0094 

0746 

0796 

0847 

0898 

51 

8.>3 

0949 

1000 

10.51 

1102 

11.53 

12W 

ia>4 

1305 

13,56 

1407 

51 

8.54 

1458 

1509 

1560 

1610 

1661 

1712 

1763 

1814 

1865 

1915 

51 

8oo 

931966 

2017 

2068 

2118 

2169 

2220 

2271 

2322 

2372 

2423 

51 

856 

2474 

2524 

2576 

2626 

2677 

2727 

2778 

2829 

2879 

2930 

51 

&57 

2981 

3031 

3082 

3133 

3183 

3234 

3286 

3335 

3386 

&137 

51 

858 

3487 

3538 

3589 

3639 

3690 

3740 

3791 

3841 

3892 

3943 

51 

859 

3993 

4044 

4094 

4145 

4195 

4216 

4296 

4347 

4397 

4448 

51 

860 

934498 

4549 

4.599 

46.50 

4700 

4751 

4801 

4852 

4902 

4953 

50 

861 

5003 

5061 

5104 

6154 

6206 

6256 

5306 

6356 

5106 

5467 

60 

862 

6507 

5558 

5608 

66.58 

6709 

6769 

6809 

6860 

5910 

5960 

50 

863 

6011 

6061 

6111 

6162 

6212 

6262 

6313 

6363 

6413 

6463 

50 

864 

6514 

6564 

6614 

6665 

6715 

6766 

6815 

6865 

6916 

6966 

60 

865 

937016 

7066 

7117 

7167 

7217 

7267 

7317 

7367 

7418 

7468 

50 

866 

7518 

7568 

7618 

7668 

7718 

7769 

7819 

7869 

7919 

7969 

50 

867 

8019 

8069 

8119 

8169 

8219 

8269 

8320 

8370 

8420 

8470 

50 

868 

8520 

mo 

8620 

8670 

8720 

8770 

8820 

8870 

8920 

8970 

50 

869 

9020 

9070 

9120 

9170 

9220 

9270 

9320 

9369 

9419 

9469 

50 

870 

939519 

9569 

9619 

9669 

9719 

9769 

9819 

9869 

9918 

9968 

50 

871 

940018 

0068 

0118 

0168 

0218 

0267 

0317 

0367 

0117 

0467 

50 

872 

0516 

a566 

0616 

0666 

0716 

0766 

0815 

0865 

0915 

1  0964 

50 

873 

1014 

1064 

1114 

1163 

1213 

1263 

1313 

1362 

1412 

1  1462 

50 

874 

1511 

1561 

1611 

1660 

1710 

1760 

1809 

1859 

1909 

1968 

60 

875 

942008 

20.58 

2107 

2167 

2207 

2266 

2306 

2355 

24a5 

2455 

50 

876 

2.501 

2.5.54 

2603 

2653 

2702 

2762 

2801 

2861 

2901 

2950 

50 

877 

3000 

3049 

3099 

3148 

3198 

3217 

3297 

3346 

3396 

3445 

49 

878 

3495 

3544 

3593 

3643 

3692 

3742 

3791 

3841 

3890 

3939 

49 

879 

3989 

4038 

4088 

4137 

4186 

4236 

4286 

4335 

4384 

4433 

49 

880 

944483 

4532 

4681 

4631 

4680 

4729 

4779 

4828 

4877 

4927 

49 

881 

4976 

502.5 

6074 

6124 

5173 

6222 

5272 

6321 

5370 

5419 

49 

882 

5469 

5518 

6567 

5616 

5666 

5715 

57M 

6813 

5862 

5912 

49 

883 

5961 

6010 

6059 

6108 

61.57 

6207 

0266 

6305 

6364 

64as 

49 

8M 

6452 

6501 

6651 

6600 

6649 

6698 

6747 

6796 

6845 

6894 

49 

88,5 

946943 

6992 

7041 

7090 

7140 

7189 

72.38 

7287 

7336 

7385 

49 

886 

74;^ 

7483 

7532 

7581 

7630 

7679 

7728 

WW 

7826 

7875 

49 

887 

7924 

7973 

8022 

8070 

8119 

8168 

8217 

8266 

8315 

8364 

49 

888 

8413 

8462 

8.511 

8560 

8(309 

86.57 

8706 

8765 

8804 

8863 

49 

889 

8902 

8961 

8999 

9048 

9097 

9146 

9196 

9214 

9292 

9341 

49 

890 

949390 

9439 

9488 

9536 

958.5 

96,34 

9683 

9731 

9780 

9829 

49 

891 

9878 

9926 

9975 

.0024 

.0073 

.0121 

.0170 

.021iJ 

.0267 

.0316 

49 

892 

950365 

0414 

0462 

Oijll 

0(560 

0608 

0(557 

0706 

!  0764 

0803 

49 

893 

0851 

0900 

0949 

0997 

1046 

1095 

1143 

1192 

1240 

1289 

49 

894 

1338 

1386 

1435 

1483 

1632 

1580 

1629 

1677 

1726 

1776 

49 

895 

961823 

1872 

1920 

1969 

2017 

2066 

2114 

2163 

2211 

2260 

48 

896 

2308 

2356 

240.5 

2453 

2.502 

2560 

2599 

2647 

2696 

2744 

48 

897 

2792 

2841 

2889 

2938 

2986 

3034 

3083 

3131 

3180 

3228 

48 

898 

3276 

3325 

3373 

3421 

3470 

3518 

3666 

3615 

3663 

3711 

48 

899 
N. 

.  3760 

3808 

3856 

3906 
3 

3953 

4001 
5 

4049 

4098 

4146 

4194 

48 
D. 

0 

1 

2 

4 

6 

7 

8 

9 

Trig.— 31. 


361 


Num.  900,  Log.  954. 

TABLE  L— LOGARITHMS 

N. 

0 

1 

2 

3 

4 

5 

6. 

7 

8 

9 

D. 

900 

954243 

4291 

4339 

4387 

4435 

4484 

4r>32 

4580 

4628 

4677 

48 

901 

4725 

4773 

4821 

4869 

4918 

4966 

5014 

5062 

5110 

6158 

48 

902 

5207 

5255 

5303 

5351 

5399 

5447 

5495 

5543 

5592 

5640 

48 

903 

5688 

5736 

5784 

5832 

5880 

5928 

5976 

6024 

6072 

6120 

48 

904 

6168 

6216 

6265 

6313 

6361 

6409 

6457 

6505 

6558 

6601 

48 

905 

956649 

6697 

6745 

6793 

6840 

6888 

6986 

6984 

7032 

7080 

48 

906 

7128 

7176 

7224 

7272 

7320 

7368 

7416 

7464 

7512 

75,59 

48 

907 

7607 

7655 

7703 

7751 

7799 

7847 

7894 

7942 

7990 

8088 

48 

908 

8086 

8134 

8181 

8229 

8277 

8325 

8373 

8421 

8468 

8516 

48 

909 

8564 

8612 

8659 

8707 

8755 

8803 

8850 

8898 

8946 

8994 

48 

910 

959041 

9089 

9137 

9185 

9232 

9280 

9828 

9875 

9423 

9471 

48 

911 

9518 

9566 

9614 

9661 

9709 

9757 

9804 

9852 

9900 

9947 

48 

912 

9995 

.0042 

.0090 

.0138 

.0185 

.0233 

.0280 

.0328 

.0376 

.0423 

48 

913 

960471 

0518 

0566 

0613 

0061 

0709 

0756 

0804 

0851 

0899 

48 

914 

0946 

0994 

1041 

1089 

1136 

1184 

1231 

1279 

1326 

1374 

47 

915 

961421 

1469 

1516 

1563 

1611 

1658 

1706 

1753 

1801 

1848 

47 

916 

1895 

1943 

1990 

2038 

2085 

2132 

2180 

2227 

2275 

2322 

47 

917 

2369 

2417 

2464 

2511 

2559 

2606 

2653 

2701 

2748 

2795 

47 

918 

2843 

^890 

2937 

2985 

8032 

3079 

8126 

3174 

3221 

3268 

47 

919 

3316 

3363 

3410 

3457 

3504 

3552 

3599 

3646 

3693 

3741 

47 

920 

963788 

38a5 

3882 

3929 

3977 

4024 

4071 

4118 

4165 

4212 

47 

921 

4260 

4307 

4354 

4401 

4448 

4495 

4542 

4590 

4637 

4684 

47 

922 

4731 

4778 

4825 

4872 

4919 

4966 

5018 

5061 

5108 

5155 

47 

923 

5202 

5249 

5296 

5343 

5390 

5437 

&484 

5531 

5578 

5625 

47 

924 

5672 

5719 

5766 

5813 

5860 

5907 

5954 

6001 

6048 

6095 

47 

925 

966142 

6189 

6236 

6283 

6829 

6876 

6428 

6470 

6517 

65&4 

47 

926 

6611 

6658 

6705 

6752 

0799 

6845 

6892 

6939 

6986 

7033 

47 

927 

7080 

7127 

7173 

7220 

7267 

7314 

7301 

7408 

7454 

7601 

47 

928 

7548 

7595 

7642 

7688 

77:35 

7782 

7829 

7875 

7922 

7969 

47 

929 

8016 

8062 

8109 

8156 

8203 

8249 

8296 

8343 

8390 

8486 

47 

930 

968483 

8530 

8576 

8623 

8670 

8716 

8768 

8810 

8856 

8903 

47 

931 

8950 

8996 

9043 

9090 

9136 

9183 

9229 

9276 

9328 

9369 

47 

932 

9416 

9463 

9509 

9556 

9602 

9649 

9695 

9742 

9789 

9835 

47 

am 

9882 

9928 

9975 

.0021 

.0068 

.0114 

.0161 

.0207 

.0254 

.0300 

47 

934 

970347 

0393 

0440 

0486 

0533 

0579 

0626 

0672 

0719 

0765 

46 

935 

970812 

0&58 

0904 

0951 

0997 

1044 

1090 

1137 

1188 

1229 

46 

936 

1276 

1322 

1369 

1415 

1461 

1508 

15,>4 

1601 

1647 

1693 

46 

937 

1740 

1786 

1832 

1879 

1925 

1971 

2018 

20(34 

2110 

2167 

46 

938 

2203 

2249 

2295 

2342 

2388 

2434 

2481 

2527 

2573 

2()19 

46 

939 

2666 

2712 

2758 

2804 

2851 

2897 

2943 

2989 

3035 

3082 

46 

940 

973128 

3174 

3220 

3266 

3313 

3359 

8405 

3451 

3497 

3543 

46 

941 

3590 

3636 

3682 

3728 

3774 

3820 

8866 

3913 

3a59 

4005 

46 

942 

4051 

4097 

4143 

4189 

4235 

4281 

4327 

4374 

4420 

4466 

46 

913 

4512 

4558 

4604 

4650 

4696 

4742 

4788 

4834 

4880 

4926 

46 

944 

4972 

5018 

5064 

5110 

5156 

5202 

5248 

5294 

5340 

5386 

46 

945 

975432 

5478 

5.524 

5570 

5616 

5662 

5707 

5753 

5799 

5845 

46 

946 

5891 

5937 

5983 

6029 

6075 

6121 

6167 

6212 

6258 

mn 

46 

947 

6a50 

6396 

6442 

6488 

6538 

6579 

6625 

6671 

6717 

6763 

46 

94e 

6808 

6854 

6900 

6946 

6992 

7037 

7083 

7129 

7175 

7220 

46 

im 

>    7266 

731ii 

7358 

7403 

7449 

7495 

7541 

7586 

7632 

7678 

46 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

362 


[ 

OF  NUMBERS. 

Hum.  999,  Log.  9S9. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

950 

977724 

7769 

7815 

7861 

7906 

79.52 

7998 

8043 

8089 

8ia5 

46 

951 

8181 

8226 

8272 

8317 

8363 

8409 

84,54 

8500 

8546 

8591 

46 

952 

8637 

8683 

8728 

8774 

8819 

8865 

8911 

8956 

9002 

9047 

46 

953 

9093 

9138 

9184 

9230 

9275 

9:521 

9366 

9412 

94.57 

9;503 

46 

954 

9548 

9594 

9639 

9685 

9730 

9776 

9821 

9867 

9912 

9958 

46 

9o5 

980003 

0049 

0094 

0140 

0185 

0231 

0276 

0322 

0367 

0412 

45 

9,56 

04.58 

0.503 

0549 

0594 

0640 

0685 

07:^ 

0776 

0821 

0867 

45 

9-57 

0912 

0957 

1003 

1048 

1093 

1139 

1184 

1229 

1275 

1320 

45 

9.58 

1366 

1411 

14,56 

1501 

1,>47 

1592 

1637 

1683 

1728 

1773 

45 

959 

1819 

1864 

1909 

1954 

2000 

2045 

2090 

2135 

2181 

2226 

45 

960 

982271 

2316 

2.362 

2407 

2452 

2497 

2.543 

2588 

2633 

2678 

45 

961 

2723 

2769 

2814 

28,59 

2904 

2949 

2994 

3040 

3085 

3130 

45 

962 

3175 

3220 

3265 

3;^io 

a'i56 

3401 

3446 

3491 

3536 

a581 

45 

96;^ 

3626 

3671 

3716 

3762 

3807 

38,52 

3897 

3942 

3987 

4032 

45 

964 

4077 

4122 

4167 

4212 

4257 

4302 

4347 

4392 

4437 

4482 

45 

965 

984,527 

4572 

4617 

4662 

4707 

4752 

4797 

4842 

4887 

4932 

45 

mi 

4977 

5022 

5067 

5112 

5157 

5202 

5247 

5292 

5337 

5382 

45 

967 

5426 

5471 

5516 

5561 

5606 

5651 

5696 

5741 

5786 

58:30 

45 

968 

5875 

5920 

596.5 

6010 

6055 

6100 

6144 

6189 

6234 

6279 

45 

969 

6324 

6369 

6413 

6458 

6503 

6548 

6593 

6637 

6682 

6727 

45 

970 

986772 

6817 

6861 

6906 

6951 

6996 

7040 

7085 

71.30 

7175 

45 

971 

7219 

7261 

7309 

7;^ 

7398 

7443 

7488 

75:^ 

7577 

7622 

45 

972 

7666 

7711 

7756 

7800 

7845 

7890 

im 

7979 

8024 

8068 

45 

973 

8113 

81,57 

8202 

8247 

8291 

83:36 

8,-i81 

8425 

8470 

8514 

45 

974 

8559 

8604 

8648 

8693 

8737 

8782 

8826 

8871 

8916 

8960 

45 

975 

9890a5 

9049 

9094 

9138 

9183 

9227 

9272 

9316 

9361 

9405 

45 

976 

94,50 

9494 

9539 

9583 

9628 

9672 

9717 

9761 

9806 

9850 

44 

977 

9895 

9939 

9983 

.0028 

.0072 

.0117 

.0161 

.0206 

.0250 

.0294 

44 

978 

990339 

om 

0428 

0472 

0,516 

0561 

0605 

06.50 

0694 

0738 

44 

979 

0783 

0827 

0871 

0916 

0960 

1004 

1049 

1093 

1137 

1182 

44 

980 

991226 

1270 

1315 

1359 

1403 

1448 

1492 

1536 

1580 

1625 

44 

981 

1669 

1713 

1758 

1802 

1846 

1890 

19,35 

1979 

2023 

2067 

44 

982 

2111 

21,56 

2200 

2244 

2288 

2333 

23-77 

2421 

2465 

2509 

44 

988 

2,554 

2598 

2642 

2686 

2730 

2774 

2819 

2863 

2907 

2951 

.  44 

984 

2995 

3039 

3083 

3127 

3172 

3216 

3260 

3304 

3348 

3392 

44 

985 

99,3436 

3480 

3524 

a568 

3613 

3657 

3701 

3745 

3789 

3833 

44 

986 

3877 

3921 

3965 

4009 

4053 

4097 

4141 

4185 

4229 

4273 

44 

987 

4317 

4361 

4405 

4449 

4493 

4537 

4581 

4625 

4669 

4713 

44 

988 

47,57 

4801 

4845 

4889 

49:« 

4977 

5021 

5065 

5108 

5152 

44 

989 

5196 

5240 

52i« 

5328 

5:^72 

5416 

5460 

5504 

5547 

5591 

44 

990 

9956.3.5 

5679 

,5723 

5767 

5811 

5854 

5898 

5942 

5986 

6030 

44 

991 

6074 

6117 

6161 

62a5 

6249 

6293 

6337 

6380 

6424 

6468 

44 

992 

6512 

6555 

6599 

6643 

6687 

6731 

6774 

6818 

6862 

6906 

44 

993 

6949 

6993 

7037 

7080 

7124 

7168 

7212 

7255 

7299 

7343 

44 

994 

7386 

7430 

7474 

7517 

7561 

7605 

7648 

7692 

7736 

7779 

44 

995 

997823 

7867 

7910 

7954 

7998 

8041 

8085 

8129 

8172 

8216 

44 

996 

8259 

8303 

8347 

8390 

8434 

8477 

8521 

8564 

8608 

8652 

44 

997 

8695 

8739 

8782 

8826 

8869 

8913 

8956 

9000 

9043 

9087 

44 

998 

9131 

9174 

9218 

9261 

9305 

9348 

9392 

9435 

9479 

9522 

43 

999 

9565 

9609 

9652 

9696 

9739 

9783 

9826 

9870 

9913 

9957 

43 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

363 


Num.  1000,  Log.  000.  TABLE  I.— LOGARITHMS 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

1000 

000000 

0043 

0087 

0130 

0174 

0217 

0260 

0304 

0347 

0391 

43 

1001 

0434 

0477 

0521 

0564 

0608 

0(i51 

0694 

0738 

0781 

0824 

43 

1002 

0868 

0911 

0954 

0998 

1041 

1084 

1128 

1171 

1214 

1268 

43 

1003 

1301 

1344 

1388 

1431 

1474 

1517 

1561 

1604 

1647 

1690 

43 

1004 

1734 

1777 

1820 

1863 

1907 

1960 

1993 

2036 

2080 

2123 

43 

1005 

002166 

2209 

22.52 

2296 

2339 

2382 

2425 

2468 

2612 

2555 

43 

1006 

2598 

2641 

2684 

2727 

2771 

2814 

2867 

2900 

2943 

2986 

43 

1007 

3029 

3073 

3116 

3159 

3202 

3245 

3288 

3331 

3374 

3417 

43 

1008 

3461 

3504 

3547 

3590 

3633 

3676 

3719 

3762 

3806 

3848 

43 

1009 

3891 

3934 

3977 

4020 

4063 

4106 

4149 

4192 

4236 

4278 

43 

1010 

004321 

4364 

4407 

4450 

4493 

4536 

4579 

4622 

4665 

4708 

43 

1011 

4751 

4794 

4837 

4880 

4923 

4966 

5009 

5052 

5096 

5138 

43 

1012 

5181 

5223 

5266 

5309 

5352 

5395 

5438 

6481 

5524 

5567 

43 

1013 

5609 

5652 

5695 

5738 

5781 

5824 

5867 

6909 

6952 

5995 

43 

1014 

6038 

6081 

6124 

6166 

6209 

6252 

6295 

6338 

6380 

6423 

43 

1015 

006466 

6509 

6552 

6594 

6637 

6680 

6723 

6765 

6808 

6851 

43 

1016 

6894 

6936 

6979 

7022 

7065 

7107 

7150 

7193 

7236 

7278 

43 

1017 

7321 

7364 

7406 

7449 

7492 

7534 

7677 

7620 

7662 

7705 

43 

1018 

7748 

7790 

7833 

7876 

7918 

7961 

8004 

8046 

8089 

8132 

43 

1019 

8174 

8217 

8259 

8302 

8345 

8387 

8430 

8472 

8515 

8558 

43 

1020 

008600 

8643 

8685 

8728 

8770 

8813 

8856 

8898 

8941 

8983 

43 

1021 

9026 

9068 

9111 

9153 

9196 

9238 

9281 

9323 

9366 

9408 

42 

1022 

9451 

9493 

9536 

9578 

9621 

9663 

9706 

9748 

9791 

9833 

42 

1023 

9876 

9918 

9961 

.0003 

.0045 

.0088 

.0130 

.0173 

.0215 

.0258 

42 

1024 

010300 

0342 

0385 

0427 

0470 

0512 

0564 

0597 

0639 

0681 

42 

1025 

010724 

0766 

0809 

0851 

0893 

0936 

0978 

1020 

1063 

1105 

42 

1026 

1147 

1190 

1232 

1274 

1317 

ia59 

1401 

1444 

1486 

1528 

42 

1027 

1570 

1613 

1655 

1697 

1740 

1782 

1824 

1866 

1909 

1951 

42 

1028 

1993 

2035 

2078 

2120 

2162 

2204 

2247 

2289 

2331 

2373 

42 

1029 

2415 

2458 

2500 

2542 

2584 

2626 

2669 

2711 

2753 

2795 

42 

1030 

012837 

2879 

2922 

2964 

3006 

3048 

3090 

3132 

3174 

3217 

42 

1031 

3259 

3301 

3343 

3385 

3427 

3469 

3511 

3553 

3596 

3638 

42 

1032 

3680 

3722 

3764 

3806 

3848 

3890 

3932 

3974 

4016 

4058 

42 

1033 

4100 

4142 

4184 

4226 

4268 

4310 

4353 

4395 

4437 

4479 

42 

1034 

4521 

4563 

4605 

4647 

4689 

4730 

4772 

4814 

4856 

4898 

42 

1035 

014940 

4982 

5024 

5066 

5108 

5150 

5192 

52.34 

5276 

5318 

42 

1036 

5360 

5402 

5444 

5485 

5,527 

6.569 

6611 

6663 

5696 

5737 

42 

1037 

5779 

5821 

5863 

5904 

5946 

5988 

6030 

6072 

6114 

6156 

42 

1038 

6197 

6239 

6281 

6323 

6365 

6407 

6448 

6490 

6532 

6574 

42 

1039 

6616 

6657 

6699 

6741 

6783 

6824 

6866 

6908 

6950 

6992 

42 

1040 

017033 

7075 

7117 

7159 

7200 

7242 

7284 

7.326 

7367 

7409 

42 

1041 

7451 

7492 

7534 

7576 

7618 

7ft59 

7701 

7743 

7784 

7826 

42 

1042 

7868 

7909 

7951 

7993 

8034 

8076 

8118 

8159 

8201 

8243 

42 

1043 

8284 

8326 

8368 

8409 

8451 

8492 

8634 

8576 

8617 

8659 

42 

1044 

8700 

8742 

8784 

8825 

8867 

8908 

8950 

8992 

9033 

9075 

42 

1045 

019116 

9158 

9199 

9241 

9282 

9324 

9366 

9407 

9449 

9490 

42 

1046 

9,53? 

9573 

9615 

9656 

9698 

9739 

9781 

9822 

9864 

9905 

42 

1047 

9947 

9988 

.0030 

.0071 

.0113 

.01.54 

.0195 

.0237 

.0278 

.0320 

41 

1048 

020361 

0403 

0444 

0486 

0527 

0568 

0610 

0661 

0693 

0734 

41 

1049 

0775 

0817 

0858 

0900 

0941 

0982 

1024 

1065 

1107 

1148 

41 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

364 


OF  NUMBERS. 

Num.  1099,  Log 

.041. 

N. 

0 

.1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

ia50 

021189 

1231 

1272 

1313 

1355 

1396 

14.37 

1479 

1520 

1561 

41 

lOol 

1603 

1644 

1685 

1727 

1768 

1809 

18,51 

1892 

1933 

1974 

41 

m2 

2016 

2a57 

2098 

2140 

2181 

2222 

226;^ 

2305 

2346 

2387 

41 

1053 

2128 

2470 

2.511 

2552 

9593 

2635 

2676 

2717 

2758 

2799 

41 

1054 

2841 

2882 

2923 

2964 

3005 

3047 

3088 

3129 

3170 

3211 

41 

1055 

023252 

3294 

333.5 

3376 

3417 

3458 

3499 

3541 

3582 

3623 

41 

ia56 

3664 

3705 

3746 

3787 

3828 

3870 

3911 

3952 

3993 

4034 

41 

1057 

4075 

4116 

41.57 

4198 

4239 

4280 

4321 

4363 

4404 

4445 

41 

ia58 

4486 

4527 

4568 

4609 

4&50 

4691 

4732 

4773 

4814 

4855 

41 

1059 

4896 

4937 

4978 

5019 

5060 

5101 

5142 

5183 

5224 

5265 

41 

1060 

025306 

5347 

5388 

5429 

5470 

5511 

5552 

5593 

5634 

5674 

41 

1001 

5715 

5756 

5797 

5&38 

5879 

5920 

5961 

6002 

6043 

6084 

41 

1062 

6125 

6165 

6206 

6247 

6288 

6329 

6370 

6411 

6452 

6492 

41 

1063 

6533 

6574 

6615 

66,56 

6697 

6737 

6778 

6819 

6860 

6901 

41 

1064 

6942 

6982 

7023 

7064 

7105 

7146 

7180 

7227 

7268 

7309 

41 

1065 

027a50 

7390 

7431 

7472 

7513 

7^i 

7594 

7635 

7676 

7716 

41 

1066 

7757 

7798 

7839 

7879 

7920 

7961 

8002 

8042 

8083 

8124 

41 

1067 

8164 

8205 

8246 

8287 

8327 

8368 

8409 

8449 

»490 

8531 

41 

1068 

8.571 

8612 

8653 

8693 

8734 

8775 

8815 

8856 

8896 

8937 

41 

1069 

8978 

9018 

9059 

9100 

9140 

9181 

9221 

9262 

9303 

9343 

41 

1070 

029384 

9424 

9465 

9506 

9.546 

9587 

9627 

9668 

9708 

9749 

41 

1071 

9789 

9830 

9871 

9911 

99.52 

9992 

.003;3 

.0073 

.0114 

.0154 

41 

1072 

030195 

0235 

0276 

0316 

0357 

0397 

(M38 

0478 

0519 

0559 

40 

1073 

0600 

0640 

0681 

0721 

0762 

0802 

0^3 

om 

0923 

0964 

40 

1074 

1004 

1045 

1085 

1126 

1166 

1206 

1247 

1287 

1328 

1368 

40 

1075 

031408 

1449 

1489 

1530 

1570 

1610 

1651 

1691 

1732 

1772 

40 

1076 

1812 

1853 

1893 

1933 

1974 

2014 

2ft->4 

2095 

2135 

2175 

40 

1077 

2216 

2256 

2296 

2337 

2377 

2417 

24.58 

2498 

2538 

2578 

40 

1078 

2619 

2659 

2699 

2740 

2780 

2820 

2800 

2901 

2941 

2981 

40 

1079 

3021 

3062 

3102 

3142 

3182 

322;3 

3263 

3303 

3343 

3384 

40 

1080 

033424 

3464 

3504 

3544 

3585 

3025 

3665 

3705 

3745 

3786 

40 

1081 

3826 

3866 

3906 

3946 

3986 

4027 

4067 

4107 

4147 

4187 

40 

1082 

4227 

4267 

4308 

4348 

4388 

4428 

4408 

4508 

4548 

4588 

40 

1083 

4628 

4669 

4709 

4749 

4789 

4829 

4869 

4909 

4949 

4989 

40 

1084 

5029 

5069 

5109 

5149 

5190 

5230 

5270 

5310 

6350 

5390 

40 

1085 

035430 

5470 

5510 

5550 

5590 

5630 

5670 

5710 

5750 

5790 

40 

1086 

5830 

5870 

5910 

5950 

5990 

6030 

6070 

6110 

6150 

6190 

40 

1087 

6230 

6269 

6309 

6349 

6389 

6429 

6469 

6509 

6549 

6589 

40 

1088 

6629 

6669 

6709 

6749 

6789 

6828 

6868 

6908 

6948 

6988 

40 

1089 

7028 

7068 

7108 

7148 

7187 

7227 

7267 

7307 

7347 

7387 

40 

1090 

037426 

7466 

7506 

7546 

7586 

7626 

7665 

7705 

7745 

7785 

40 

1091 

782) 

7865 

7904 

7944 

7984 

8024 

8064 

8103 

8143 

8183 

40 

1092 

8223 

8262 

8302 

8342 

8382 

&421 

8461 

8501 

8541 

8580 

40 

1093 

8620 

8660 

8700 

8739 

8779 

8819 

8859 

8898 

8938 

8978 

40 

1094 

9017 

9057 

9097 

9136 

9176 

9216 

9255 

9295 

9335 

9374 

40 

1095 

039414 

9454 

9493 

9533 

9573 

9612 

9652 

9692 

9731 

9771 

40 

1096 

9811 

9850 

9890 

9929 

9969 

.0009 

.0048 

.0088 

.0127 

.0167 

40 

1097 

040207 

0246 

0286 

0325 

0365 

0405 

0444 

0484 

0523 

0563 

40 

1098 

0602 

0642 

0681 

0721 

0761 

0800 

0840 

0879 

0919 

0958 

40 

1099 

0998 

1037 

1077 

1116 

1156 

1195 

1235 

1274 

1314 

1353 

39 

N. 

0 

1 

2 

8 

4 

6 

6 

7 

8 

9 

D. 

365 


TABLE 

IL— LOGARITHMS  OF 

PRIME           j 

N. 

Logarithm. 

N. 

Logarithm. 

N. 

Logarithm, 

2 

30102  99956  63981 

238 

367;i5  59210  26019 

547 

73798  73263  33431 

3 

47712  12.547  19662 

239 

37839  79009  48138 

557 

74585  51951  73729 

5 

69897  00043  36019 

241 

38201  70425  74868 

563 

75050  83948  51346 

7 

84509  80400  14257 

251 

39967  37214  81038 

569 

75511  22663  95071 

11 

04139  26851  58225 

257 

40993  31233  31295 

571 

75663  61082  45848 

13 

11394  33523  06837 

263 

41995  57484  89758 

577 

76117  58131  55731 

17 

23044  89213  78274 

269 

42975  22800  02408 

587 

76863  81012  47614 

19 

27875  36009  52829 

271 

43296  92908  74406 

593 

77305  469.33  64263 

23 

36172  78360  17593 

277 

44247  97690  64449 

599 

77742  68223  89311 

29 

46239  79978  98956 

281 

44870  63199  05080 

601 

77887  44720  02740 

31 

49136  16938  34273 

283 

45178  64355  24290 

607 

78318  86910  75258 

37 

56820  17210  66995 

293 

46686  76203  54109 

613 

78746  04745  18415 

41 

61278  38567  19735 

307 

48713  837.54  77186 

617 

79028  51640  33242 

43 

63346  84555  79587 

311 

49276  03890  26838 

619 

79169  06490  20118 

47 

67209  78579  35717 

313 

49554  43375  46448 

631 

80002  93592  44134 

53 

72427  58696  00789 

317 

50ia5  92622  17751 

641 

80685  80295  18817 

59 

77085  20116  42144 

331 

51982  79937  75719 

643 

80821  09729  24222 

61 

78532  98850  10767 

337 

52762  99008  71339 

647 

81090  42806  68700 

67 

82607  48027  00826 

847 

54032  94747  90874 

653 

81491  31 812  75074 

71 

85125  83487  19075 

349 

54282  54269  59180 

659 

81888  54145  94010 

73 

86332  28601  20456 

353 

54777  470.53  87823 

661 

82020  14594  &5640 

79 

89762  70912  90441 

359 

55509  44485  78319 

673 

82801  50642  23977 

83 

91907  80923  76074 

367 

56466  60642  52089 

677 

83058  86686  85144 

89 

94939  00066  44913 

373 

57170  88318  08688 

683 

83442  07036  81533 

97 

98677  17342  66245 

379 

57863  92099  68072 

691 

83947  80473  74198 

101 

00432  13737  82643 

383 

58319  87739  68623 

701 

84571  80179  66659 

103 

01283  72247  05172 

389 

58994  96013  25708 

709 

85064  62351  83067 

107 

02938  37776  85210 

397 

59879  05067  63115 

719 

85672  88903  82883 

109 

03742  64979  40624 

401 

60314  43726  20182 

727 

86153  44108  59038 

113 

05307  84434  83420 

409 

61172  33080  07342 

733 

86510  39746  41128 

127 

10380  37209  55957 

419 

62221  40229  66295 

739 

86864  44383  94826 

131 

11727  12956  55764 

421 

62428  20958  35668 

743 

87098  88137  60575 

137 

13672  05671  56407 

431 

63447  72701  60732 

751 

87563  99370  04168 

139 

14301  48002  54095 

433 

63648  78963  53365 

757 

87909  58795  00073 

149 

17318  62684  12274 

439 

64246  45202  42121 

761 

88138  46567  70573 

151 

17897  69472  93169 

443 

64640  37262  23070 

769 

88592  63398  01431 

157 

19589  9ft524  09234 

449 

65224  63410  03323 

773 

88817  94939  18325 

163 

21218  76044  03958 

457 

65991  62000  69850 

787 

89597  47323  59065 

167 

22271  64711  47583 

461 

66370  092,53  89648 

797 

90145  83213  96112 

173 

23804  61031  28795 

463 

66558  09910  17953 

809 

90794  85216  12272 

179 

25285  30309  79893 

467 

66931  68805  66112 

811 

90902  0a542  111.56 

181 

25767  85748  69185 

479 

68033  55134  14563 

821 

91434  31571  19441. 

191 

28103  33672  47728 

487 

68752  89612  14634 

823 

91539  mm  12270 

193 

28555  73090  07774 

491 

69108  14921  22968 

827 

91750  55095  52547 

197 

29446  62261  61593 

499 

69810  05456  23390 

829 

91855  45305  50274 

199 

29885  30764  09707 

503 

70156  79850  55927 

839 

92376  19608  28700 

211 

32428  24.552  97693 

509 

70671  77823  36759 

853 

93094  90311  67523 

223 

34830  48630  48161 

521 

71683  77232  99524 

857 

93298  08219  23198 

227 

35602  58571  93123 

523 

71850  16888  67274 

859 

93399  316;i8  31242 

229 

35983  54823  39888 

541 

73319  72651  06569 

863 

93601  07957  15210 

366 


NUMBERS  LESS  THAN  1000. 


877 
881 
883 
887 
907 
911 


Logarithm. 

N. 

94299  959a3  66041 

919 

94497  59084  12048 

929 

94596  070a5  77r)69 

937 

94792  36198  31726 

941 

95760  72870  60095 

947 

95951  83769  72998 

9.53 

Logarithm. 

TS. 

mSSl  5.5113  861 11 

967 

96801  57139  93642 

971 

97173  95908  87778 

977 

97^58  962;M  272.57 

983 

976;^  99790  03273 

991 

97909  29006  38326 

997 

Logarithm. 


98,542  64740  .83002 
98721  92299  08005 
98989  456;S7  18773 
992.5.5  35178  .32136 
99607  36.544  8.5275 
99869  51583  11656 


In  the  above  table,  only  the  mantissas  are  given ;  the 
characteristics  may  be  found  by  the  rule  (908). 

By  means  of  these  logarithms,  the  logarithm  of  any 
number  may  be  found  with  equal  accuracy.  If  the  given 
number  be  the  product  of  any  of  the  prime  numbers  in 
the  table,  its  logarithm  may  be  found  by  addition  (912). 
For  example, 

log.  6  =  log.  2  + log.  3==    .77815  12503  83G43; 
log.  1001=  log.  7  + log.  11  + log.  13  =  3.00043  40774  79319. 

These  results  may  err  in  the  last  figure ;  the  loga- 
rithm of  6  to  fifteen  figures,  has  the  last  figure  nearer  to 
4  than  to  3. 

When  the  given  number  is  not  the  product  of  numbers 
in  the  table,  its  logarithm  may  be  calculated  by  the  fol- 
lowing formulas : 


M 


.43429  44819  0325; 
log.  n  =  log.  (n-l)  +  2M  (.y^^  + 


1 


3(2  n  — ly 


+  &C 


Omitting  the  second  fraction  in  the  parenthesis,  the 
logarithm  will  be  found  correct  to  three  times  as  many 
figures  as  there  are  in  the  number  n.  Using  this  term 
gives  the  result  true  to  five  times  as  many  figures  as  there 
are  in  n.     For  example,  to  find  the  logarithm  of  1013, 

log.  1012  =  2  log.  2  +  log.  11  +log.  23  =  3.00518  05125  03780 
2M-J-2025  =   .00042  89328  21633 

2  M -i- 3(2025)3  =   .00000  00000  34867 

log.  1013  =3.00560  94453  60280 

For  some  large  numbers  it  may  be  necessary  to  repeat 
the  operation.  When  one  of  the  prime  factors  oi  n  —  1 
is  greater  than  1000,  it  may  be  better  to  find  the  loga- 
rithm of  n  +  1,  and  then  log.  n  by  subtracting  the  differ- 
ence. For  example,  log.  2027  can  be  found  more  readily 
from  log.  2028  than  from  log.  2026. 

367 


TABLE  IIL- 

-NATURAL  SINES. 

Deg. 

0' 

10' 

20' 

30' 

40' 

50' 

60' 

Deg. 

0 

000000 

002909 

00.5818 

008727 

011635 

014.544 

017452 

89 

1 

017452 

020361 

02)269 

020177 

02908.5 

031992 

034899 

88 

2 

031899 

037806 

040713 

043619 

04ft325 

049431 

052336 

87 

3 

052;336 

055211 

a58145 

061049 

0639.32 

066854 

069750 

86 

4 

069756 

072658 

075559 

078459 

081359 

084258 

0871.50 

85 

5 

087156 

090053 

0929.50 

095846 

098741 

101635 

104528 

84 

6 

104528 

107421 

110313 

113203 

110093 

118982 

121809 

83 

7 

121869 

124756 

127642 

130526 

133410 

136292 

139173 

82 

8 

139173 

142053 

144932 

147809 

150686 

L53561 

156434 

81 

9 

156434 

159307 

162178 

165048 

167916 

170783 

173648 

80 

10 

173648 

176512 

179375 

182236 

185095 

187953 

190809 

79 

11 

190809 

193661 

19&"j17 

199368 

202218 

205065 

207912 

78 

12 

207912 

210756 

21^3599 

216440 

219279 

222116 

224951 

77 

13 

221951 

227784 

230616 

233445 

236273 

239098 

241922 

76 

14 

241922 

244743 

247563 

250380 

253195 

256008 

258819 

75 

15 

258819 

261628 

264434 

267238 

270040 

272840 

27.5637 

74 

16 

275637 

278432 

281225 

284015 

286803 

289589 

292372 

73 

17 

292372 

295152 

297930 

300706 

303479 

306249 

309017 

72 

18 

309017 

311782 

314">45 

317305 

320062 

322816 

325568 

71 

19 

325.568 

328317 

331063 

333807 

336547 

339285 

342020 

70 

20 

342020 

344752 

347481 

350207 

352931 

355651 

358368 

69 

21 

a58368 

361082 

363793 

366501 

3(59206 

371908 

374(,07 

68 

22 

374607 

377302 

379994 

382683 

385369 

388052 

3007.31 

67 

23 

390731 

393407 

39G080 

398749 

401415 

404078 

400737 

66 

24 

406737 

409392 

412045 

414693 

417338 

419980 

422618 

65 

25 

422618 

42525:3 

427884 

430511 

4331.35 

435755 

438371 

64 

26 

438371 

440984 

443593 

44C198 

448799 

451397 

453990 

63 

27 

453990 

456i580 

459166 

461749 

464327 

466901 

469472 

62 

28 

469172 

472038 

474600 

477159 

479713 

482263 

484810 

61 

29 

484810 

487^52 

489890 

492424 

494953 

497479 

500000 

60 

30 

500000 

502517 

505030 

507538 

510043 

512543 

5150.38 

69 

31 

51.3038 

517529 

520016 

522499 

524977 

527450 

529919 

58 

32 

529919 

5.32384 

534844 

537.300 

539751 

5-12197 

544639 

57 

,33 

&44639 

517076 

519.509 

551937 

&54360 

556779 

559193 

56 

34 

559193 

561602 

564007 

566406 

668801 

571191 

573576 

55 

a5 

573576 

5759o7 

5783.32 

580703 

583069 

585429 

687785 

.54 

36 

5877&5 

590136 

592482 

594823 

597159 

599489 

601815 

53 

37 

601815 

604136 

606451 

608761 

611067 

613367 

615661 

52 

38 

61.5661 

617951 

62023;) 

622515 

624789 

627057 

629320 

51 

39 

629320 

631578 

633831 

636078 

638320 

640557 

642788 

50 

40 

642788 

645013 

647233 

649448 

6.51657 

65^3861 

656059 

49 

41 

6.560.59 

ft58252 

6604.39 

6626^0 

664796 

666966 

669131 

48 

42 

669131 

671289 

673143 

67.S590 

677732 

679868 

681998 

47 

43 

681998 

681123 

686242 

688:355 

690462 

692563 

6^4658 

46 

44 

694658 

696748 

698832 

700909 

702981 

705047 

707107 

45 

Deg. 

60' 

50' 

40' 

30' 

20' 

10' 

0' 

Deg. 

NATUI 

lAL  CO 

SINES. 

368 


TABLE 

III.— NATURAL  TANGENTS. 

1 

1 

Deg. 

0' 

icy 

2^ 

30' 

40' 

50' 

60' 

Deg.  ! 

0 

000000 

002909 

0a5818 

008727 

0116.36 

014545 

017455 

89 

1 

0174.5.5 

020:36.5 

023275 

026186 

029097 

032009 

034921 

88 

2 

034921 

037834 

040747 

04:3661 

046576 

019491 

052408 

87 

3 

052108 

a55325 

058243 

061163 

064083 

067004 

0()9927 

86 

4 

069927 

072851 

075775 

078702 

081629 

084558 

087489 

85 

5 

087489 

090421 

0933.54 

096289 

099226 

102104 

105104 

84 

6 

105104 

108046 

110990 

113936 

116883 

119833 

122785 

83 

7 

122785 

12.5738 

128694 

131652 

134613 

137.576 

140541 

82 

8 

140.541 

143508 

146478 

149451 

152426 

155404 

158384 

81 

9 

158384 

161368 

164354 

167343 

170331 

173329 

176327 

80 

10 

176.327 

179328 

18^3^ 

185339 

188^49 

191.363 

194380 

79 

11 

194380 

197401 

200425 

20ai52 

206483 

209518 

212557 

78 

12 

212.557 

215599 

218645 

221695 

224748 

227806 

230868 

77 

13 

2.30868 

2339.34 

237004 

240079 

243157 

246241 

249328 

76 

14 

249328 

252420 

255516 

258618 

261723 

204834 

267949 

75 

15 

267949 

271069 

274194 

277325 

280460 

283600 

286745 

74 

16 

286745 

289896 

293052 

296213 

299380 

302553 

305731 

73 

17 

305731 

308914 

312104 

315299 

3ia500 

321707 

324920 

72 

18 

324920 

328139 

331.364 

331595 

a37833 

»41077 

^44328 

71 

19 

344328 

34758.5 

350848 

354119 

357396 

360679 

363970 

70 

20 

363970 

367268 

370573 

373885 

877204 

380530 

383804 

69 

21 

383864 

3^205 

390.554 

393910 

397275 

400046 

401026 

68 

22 

404026 

407414 

410810 

414214 

417626 

421046 

424475 

67 

23 

424475 

427912 

43i:i58 

4»4812 

438276 

441748 

445229 

66 

24 

445229 

448719 

452218 

455726 

459244 

462771 

466308 

65 

2.5 

466308 

469854 

473410 

476976 

480551 

484137 

487733 

64 

26 

48773;^ 

491:339 

4949.5i5 

498582 

502219 

505867 

509525 

63 

27 

50952.5 

513195 

516875 

620.567 

624270 

627984 

531709 

62 

28 

5;J1709 

6:35146 

539195 

5429.56 

646728 

550513 

554309 

61 

29 

5M309 

558118 

561939 

665773 

669619 

573478 

577350 

60 

30 

577*50 

581235 

685134 

589045 

592970 

696908 

600861 

59 

31 

600861 

604827 

608807 

612801 

616809 

620832 

624869 

68 

32 

024869 

628921 

632988 

637070 

041167 

645280 

649408 

67 

m 

649408 

6.53.551 

657710 

661886 

666077 

670284 

674509 

66 

34 

674.509 

678749 

683007 

687281 

691572 

695881 

700208 

65 

a5 

700208 

7045.51 

708913 

713293 

717691 

722108 

726543 

64 

36 

720.543 

730996 

7.35469 

739961 

744472 

749003 

753554 

53 

37 

7,53551 

75812-5 

762716 

767327 

771959 

T76612 

781286 

52 

38 

781286 

785981 

790697 

795436 

800196 

804979 

809784 

61 

39 

809784 

'  814612 

819463 

824:336 

829234 

834155 

839100 

50 

40 

839100 

844069 

849062 

8.54081 

859124 

864193 

809287 

49 

41 

869287 

874407 

879.553 

884725 

889924 

895151 

900404 

48 

42 

900101 

905685 

910994 

916331 

921697 

927091 

932515 

47 

43 

9.32515 

937968 

943451 

948965 

951508 

960083 

965689 

46 

44 

965689 

971326 

976996 

982697 

988432 

994199 

1.000000 

45 

Deg. 

60' 

50' 

40' 

30' 

20' 

10' 

O' 

Deg. 

N 

ATURA 

L  COTA 

NGENT 

3. 



369 


1 
j 

TABLE  III.- 

-NATURAL  SINES. 

1 

f 

Deg. 

0' 

10' 

20' 

30' 

40' 

50' 

60' 

Deg. 

45 

707107 

709161 

711209 

713250 

715286 

717316 

719340 

44 

46 

719340 

721.3.57 

723369 

725374 

727374 

729307 

731354 

43 

47 

7313.54 

733334 

7^5309 

737277 

739239 

741195 

743145 

42 

48 

743145 

745088 

747025 

748956 

750880 

752798 

754710 

41   i 

49 

754710 

756615 

758514 

760406 

762292 

764171 

766044 

40   1 

50 

766044 

767911 

769771 

771625 

773472 

775312 

777146 

39  : 

51 

777146 

778973 

780794 

782608 

784416 

786217 

788011 

38   1 

52 

788011 

789798 

791579 

793a53 

795121 

796882 

798636 

37   ! 

53 

798636 

800383 

802123 

8038.57 

805584 

807304 

809017 

36 

54 

809017 

810723 

812423 

814116 

815801 

817480 

819152 

35 

55 

819152 

820817 

822475 

824126 

825770 

827407 

829038 

34 

56 

829038 

830661 

832277 

833886 

8a5488 

837083 

838671 

33 

57 

838671 

840251 

841825 

843391 

844951 

846503 

848048 

32 

58 

848048 

849586 

851117 

852640 

854156 

855665 

857167 

31 

59 

857167 

858662 

860149 

861629 

863102 

864567 

866025 

30 

60 

866025 

867476 

868920 

870356 

871784 

873206 

874620 

29 

61 

"  874620 

876026 

877425 

878817 

880201 

881578 

882948 

28 

62 

882948 

8S4309 

88.5664 

887011 

888350 

889682 

891007 

27 

63 

891007 

892323 

893633 

894934 

896229 

897515 

898794 

26 

64 

898794 

900065 

901329 

902585 

903834 

905075 

906308 

25 

65 

906308 

907533 

908751 

909961 

911164 

912358 

913545 

24 

66 

913545 

914725 

915896 

917060 

918216 

919364 

920505 

23 

67 

920505 

921638 

922762 

923880 

924989 

926090 

927184 

22 

68 

927184 

928270 

929348 

930418 

931480 

932534 

933580 

21 

69 

933580 

934619 

935650 

936672 

937687 

938694 

939693 

20 

70 

939693 

9406S4 

941666 

942641 

943609 

944568 

945519 

19 

71 

945^319 

946462 

947397 

948324 

949243 

950154 

951057 

18 

72 

951057 

951951 

952838 

953717 

954.588 

955450 

956305 

17 

73 

9.56305 

9.57151 

957990 

958820 

959642 

960456 

961262 

16 

74 

961262 

962059 

962849 

963630 

964404 

965169 

965926 

15 

75 

965926 

966675 

967415 

968148 

968872 

969588 

970296 

14 

76 

970296 

970995 

971687 

972370 

973045 

973712 

974370 

13 

77 

974370 

975020 

975662 

976296 

976921 

977539 

978148 

12 

78 

978148 

978748 

979341 

97992.5 

98a500 

981068 

981627 

11 

79 

981627 

982178 

982721 

983255 

983781 

984298 

984808 

10 

80 

984808 

985309 

985801 

986286 

986762 

987229 

987688 

9 

81 

987688 

988139 

988582 

989016 

989442 

989859 

990268 

8 

82 

990268 

990669 

991061 

991445 

991820 

992187 

992546 

7 

83 

992546 

992896 

993238 

993572 

993897 

994214 

994522 

6 

84 

994522 

994822 

995113 

995396 

995671 

995937 

996195 

5 

85 

996195 

996444 

9966g5 

996917 

997141 

997357 

997564 

4 

86 

997564 

99776;^ 

997953 

998135 

998308 

998473 

998630 

3 

87 

998630 

998778 

998917 

999048 

999171 

999285 

999391 

2 

88 

999.391 

999488 

999577 

999657 

999729 

999793 

999848 

1 

89 

999848 

999894 

999932 

999962 

999983 

999996 

1.000000 

0 

Deg. 

60' 

50' 

40' 

30' 

20' 

10' 

0' 

Deg. 

NATUB 

.AL  CO 

SINES. 

370 


Fj 

TABLE  IIL— NATURAL  TANGENTS. 

Deg. 

0' 

10' 

20' 

30' 

40' 

50' 

60' 

Deg. 

45 

1.000000 

1.005835 

1.011704 

1.017607 

1.023546 

1.029.520 

1.0a>530 

44 

46 

1.035530 

1.041577 

1.047060 

1.0.53780 

1.059938 

l.W)6134 

1.072369 

43 

47 

1.072:369 

1.078642 

1.0849.55 

1.091309 

1.0977021 

1.1041:37 

1.110612 

42 

48 

1.110612 

1.117131 

1.123691 

l.l;%294 

1.1:36941 

1.14:3633 

1.1.50.368 

41 

49 

1.150368 

1.157149 

1.163976 

1.170850 

1.177770 

1.184738 

1.191754 

40 

50 

1.191754 

1.198818 

1.205933 

1.213097 

1.220312 

1.227579 

1.234897 

39 

51 

1.231897 

1.212269 

1.249693 

1.2.57172 

1.261706 

1.272296 

1.279W2 

38 

52 

1.279912 

1.287645 

1.29.5106 

1.303225 

1.311105 

1.319014 

1.. 327045 

37 

53 

1.327045 

l.;«5108 

1.34.3233 

l.a51422 

l.a59676 

1.367996 

1.376:382 

36 

54 

1.376382 

1.3»1835 

1.393357 

1.401948 

1.410610 

1.419343 

1.428148 

35 

55 

1.428148 

1.437027 

1.44.5980 

1.455009 

1.464115 

1.473298 

1.482561 

34 

56 

1.482561 

1.491904 

1.501328 

1.510835 

1.. 520426 

1..530102 

1.539865 

33 

57 

1.539805 

1.549716 

1..5;396.>5 

1.569686 

1.579808 

l.,590024 

1.600335 

32 

58 

1.600a3.j 

1.610742 

1.621247 

l.&3ia52 

1.642.558 

1.65.3366 

1.664279 

31 

59 

1.664279 

1.675299 

1.686426 

1.69766^3 

1.709012 

1.720474 

1.732051 

30 

60 

1.732051 

1.743745 

1.75.55.59 

1.767494 

1.7795.52 

1.791736 

1.804048 

29 

61 

1.804048 

1.816189 

1.829063 

1.^1771 

1.854616 

1.867600 

1.880726 

28 

62 

1.880726 

1.893997 

1.907415 

1.920982 

1.9.S4702 

1.948577 

1.962611 

27 

63 

1.962611 

1.976805 

1.991164 

2.005690 

2.020:380 

2.0.3-52-56 

2.050304 

26 

64 

2.050304 

2.063532 

2.080944 

2.096544 

2.1123:35 

2.128321 

2.144507 

25 

65 

2.144507 

2.160896 

2.177492 

2.194300 

2.211323 

2.228.568 

2.246037 

24 

66 

2.246037 

2.263736 

2.281609 

2. 29984:3 

2.318261 

2.336929 

2.a55852 

23 

67 

2.35.5852 

2.37i5a37 

2.394489 

2.414214 

2.4,34217 

2.454,506 

2.475087 

22 

68 

2.475087 

2.495966 

2.517151 

2.-5:38648 

2.  .560465 

2.582609 

2.C05089 

21 

69 

2.605089 

2.627912 

2.651087 

2.674621 

2.698525 

2.722808 

2.747477 

20 

70 

2.747477 

2.772515 

2.798020 

2.823913 

2.850235 

2.876997 

2.904211 

19 

71 

2.904211 

2.931888 

2.960042 

2.9886a5 

3.017830 

3.047492 

3.077684 

18 

72 

3.077684 

3.108421 

3.139719 

3.171.5a5 

3.204064 

3.237144 

3.27085:3 

17 

73 

3.2708.>3 

3.305209 

3.3402:53 

3.375943 

3.4123«j:3 

3.449512 

3.487414 

16 

74 

3.487414 

3.526094 

3.565575 

3.6058&1 

3.647047 

3.689093 

3.732061 

15 

75 

3.732051 

3.775952 

3.820828 

3.866713 

3.913642 

3.961652 

4.010781 

14 

76 

4.010781 

4.061070 

4.112561 

4.165300 

4.219a32 

4.274707 

4.331476 

13 

77 

4.331476 

4.389694 

4.449418 

4.510709 

4..57:3629 

4.638246 

4.704630 

12 

78 

4.704630 

4.7728.57 

4.843005 

4.9151.57 

4.989403 

6. 0(5.58:35 

5.144554 

11 

79 

5.144554 

5.225665 

6.309279 

6.395517 

6.484505 

6.576379 

5.671282 

10 

80 

5.671282 

5.769369 

5.870804 

5.975764 

6.0844.38 

6.197028 

6.313752 

9 

81 

6.313732 

6.434843 

6.. 560.554 

6.69115() 

6.820944 

6.968234 

7.11.5.370 

8 

82 

7.115370 

7.26872.5 

7.428706 

7.5957.54 

7.770:351 

7.9.53022 

8.144346 

7 

83 

8.144346 

8.344956 

8.-55.5547 

8.77(5887 

9.009826 

9.255.304 

9..ai.%4 

6 

84 

9.514364 

9.788173 

10.07803 

10.3854C 

10.71191 

11.05943 

11.43005 

6 

85 

11.43005 

11.82617 

12.2.50,51 

12.70620 

13.19688 

13.72674 

14.30067 

4 

86 

14.30067 

14.92442 

15.60478 

16.34986 

17.16934 

18.07498 

19.08114 

3 

87 

19.08114 

20.20.355 

21.47040 

22.90377 

24.54176 

26.43160 

28.6362.5 

2 

88 

28.63625 

31.241.58 

34.36777 

38.18846 

42.96408 

49.10:388 

57.28996 

1 

89 

57.28996 

68.75009 

85.93979 

114.5887 

171.8864 

343.7737 

00 

0 

Deg. 

60' 

50' 

40' 

30' 

20' 

10' 

0' 

Deg. 

NATURAL  COTANGENTS. 

371 


M. 

0 

1 

2 
3 

4 
5 
6 
.  7 
8 
9 
10 
11 
12 
13 
U 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
2-3 
26 
27 
28 
29 
30 
31 
32 
33 
34 
So 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

\~m7 

89 '^ 


TABLE   IV.— LOGARITHMIC 


6.463726 
764756 
910847 

7.065786 
162696 
241877 


366816 
417968 
463726 

7.505118 
542906 
577668 
609853 
639816 
667845 
694173 
718997 
742478 
764754 

7.785943 
806146 
825451 
843934 
861662 


895085 
910879 
92(3119 
940842 

7.95.5082 
968870 
982233 
995198 

8.007787 
020021 
031919 
W3301 
0.54781 
065776 

8.076500 
08C9 
097183 
107107 
110926 
126471 
i;35810 
144953 
irMQiJ 
162681 

8.171280 
179713 
187985 
196102 
204070 
211895 
219581 
227134 
234557 
24185.5 


5017 
2934 
2082 
1615 
1319 
1115 
5 

852.5 
762.6 
689.8 
629.8 
579.3 
536.4 
499.3 
467.1 
438.8 
413.7 
391.3 
371.2 
353.1 
336.7 
321.7 
308.0 
295.4 
283.9 
273.2 
263.2 
254.0 
245.3 
237.3 
229.8 
222.7 
216.1 
209.8 
203.9 
198.3 
193.0 
188.0 

ia3.2 

178.7 
174.4 
170.3 
166.4 
162.6 
159.1 
155.6 
152.4 
149.2 
146.2 
143.3 
140.5 
137.8 
135.3 
132.8 
130.4 
128.1 
125.9 
123.7 
121.6 


Tang.  PPl"  M^ 
60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 


6.463726 
764756 
940847 

7.065786 
162696 
241878 
30882;5 
366817 
417970 
463727 

7.505120 
542909 
577672 
609857 
639820 
667849 
694179 
719003 
742484 
764761 

7.7859,51 
8061.5.5 
82.5460 
843944 
861674 
878708 


910894 

926134 

940858 

7.955100 


9822.53 
995219 

8.007809 
020044 
031945 
01:3.527 
051809 
085800 

8.076531 
086997 
097217 
107203 
110963 
126510 
1358.51 
144996 
153952 
162727 

8.171328 
179763 
188036 
1961.56 
204126 
211ft53 
219641 
227195 
234621 
241921 


5017 
293.5 
2082 
1615 
1320 
1116 
966.5 
852.5 
762.6 
689.9 
629.8 
579.4 
536.4 
499.4 
467.1 
438.8 
413.7 
391.3 
371.2 
353.2 
336.7 
321.7 
308.0 
295.5 
283.9 
273.2 
.2 
2.54.0 
245.4 
237.3 
229.8 
222.7 
216.1 
209.8 
203.9 
198.3 
193.0 
188.0 
183.3 
178.7 
174.4 
170.3 
166.4 
162.7 
1.59.1 
1.55.7 
152.4 
149.3 
146.2 
143.3 
140.6 
137.9 
135.3 
132.8 
130.4 
128.1 
125.9 
123.8 
121.7 


Cosine.  PPl"  (lotans.  PPl"  M 


Sin-. 
8.241855 
249033 
256094 
263042 


276614 
283243 
289773 
296207 
302546 
308794 

8.314954 
321027 
327016 
332924 
338753 
344504 
350181 
355783 
361815 
366777 

8.372171 
377499 
3S2762 
387962 
393101 
398179 
403199 
408161 
41,3068 
417919 

8.42271 
427462 
432156 
436800 
441394 
445941 
450440 
454893 
459301 
463665 

8.467985 
472263 
476498 
480693 
484848 
488963 
493040 
497078 
501080 
505045 

8.508974 
512867 
516726 
52a551 
524343 
528102 
531828 
535523 
539186 
512819 


Cosinf 


PIM" 

119.(; 

117.7 

115.8 

114.0 

112.2 

110.5 

108.8 

107.2 

105.6 

104.1 

102.7 

101.2 

99.82 

98.47 

97.14 

95.86 

94.60 

93.38 

92.19 

91.03 

89.90 

88.80 

87.72 

86.6 

85.64 

84. 

83. 

82.71 

81.77 

80. 

79. 

79.09 

78.23 

77.40 

76.57 

75.77 

74.99 

74.22 

73.46 

72.73 

72.00 

71.29 

70.60 

69.91 

69.24 

68.59 

67.94 

67.31 

66.69 

66.08 

65.48 

64. 

64.31 

63.75 

63.19 

62.64 

62.11 

61.58 

61.06 

60.55 


PPl' 


Tang. 


8.241921 
219102 
a561ft-) 
263115 
2699,56 
276691 
283323 
289856 
296292 
3026J34 
308884 

8.315046 
321122 
327114 
33302,5 
3388,50 
344610 
3502891 
3558951 
3614301 


PPl" 


8.3722t)2 
377622 
382889 
388092 
893234 
398315 
403338 
408304 
413213 
418068 

8.422869 
427618 
432315 
436962 
441.560 
446110 
450613 
455070 
459481 
463849 

8.468172 
472454 
476693 
480892 
4&5050 
489170 
493250 
497293 
501298 
505267 

8.509200 
513098 
516961 
520790 
524586 
528349 
532080 
535779 
539447 
543084 


119.7 
117.7 
115.8 
114.0 
112.2 
110.5 
108.9 
107.2 
10,5.7 
104.2 
102.7 
101.3 
99.87 
98.51 
97.19 
195.90 
94.65 
93.43 
92.24 
91.08 
89.95 
88.85 
87.77 
86.72 
85.70 
84.70 
83.71 
82.76 
81.82 
80.91 
80.02 
79.14 
78.29 
77.45 
76.63 
75.83 
75.05 
74.28 
73.52 
72.79 
72. 
71.35 
70.66 
69.98 
69.31 
68.65 
68.01 
67.38 
66.76 
66.15 
65.55 
64.96 
64.39 
03.82 
63.26 
62.72 
62.18 
61.65 
61.13 
60.62 


Cotiniff. 


PPl" 


H72 


HS^ 


2" 


SINES  AND  TANGENTS. 


3« 


87" 


Sine.      IPPl"      Tang.      FPl"    M. 


8.512819 
516422 
M9995 
553.389 
557054 
560540 
563999 
567431 


574214 
577566 


60.04 
59.  .55 
59.06 
58.58 
.58.11 
.57.65 
.57.19 
.56.74 
56.30 
55.87 


8.580892  f.-^i 
584193   " 


587469 
590721 


5971.52 
600332 


609734 
8.612823 
615891 
618937 
621962 
624965 
627948 
630911 
63:^854 
636776 
639680 
8.642563 
64.5428 
648274 
651102 
653911 
65G702 
659475 
662230 
664968 
6671 
8.670.393 
673080 
675751 
678405 
6S1043 
683665 
686272 
688803 
691438 


8.698.543 
699073 
701589 
70;090 
706.577 
709049 
711.507 
7139.52 
716:}8;3 
718800 


,54.60 
;54.19 
53.79 
.5:^.39 
53.00 
52.61 
52.23 
51.86 
.51.49 
,51.12 
50.76 
50.41 
,50.06 
49.72 
49.-38 
49.04 
48.71 
48.39 
48.06 
47.75 
47.43 
47.12 
40.82 
46.. 52 
46.22 
4;5.92 
4.5.6:3 
45.  a5 
45.06 
44.79 
U.'A 
44.24 
4.3.97 
43.70 
43.44 
43.18 
42.92 
42.67 
42.42 
42.17 
41.92 
11.68 
41.44 
41.21 
40.97 
40.74 
40.51 
40.29 


Cosinf 


PPI 


8..54;»84 
.546691 
550268 
55,3817 
557a3() 
560828 
564291 
567727 
571l;37 
574.520 
577877 

8.581208 
584514 
587795 
59ia51 
594283 
597492 
600677 


610094 
8.613189 
616262 
619313 
622.343 
62.5a52 
628340 
631308 
6342.56 
637184 
640093 
8.642982 
6458-53 
648704 
651.537 
654352 
657149 
659928 


()6543;3 
668160 
.670870 
67:3563 
6702:39 
678900 
681.544 
684172 
686784 


69196:3 
694529 
8.697081 
699617 
702139 
704^40 
707140 
709618 
71208:3 
7145^4 
716972 
719396 


Cotang. 


60.12 

59.62 

59.14 

58.66 

58.19 

57.73 

57.27 

56.82 

56.38 

.55.95 

55.52 

55.10 

.54.68 

.54.27 

.53.87 

53.47 

53.08 

52.70 

52.32 

51.94 

51.58 

51.21 

50.85 

50.50 

.50.15 

49.81 

49.47 

49.13 

48.80 

48.48 

48.16 

47.^4 

47..53 

47.22 

46.91 

46.61 

46.31 

46.02 

4.5.73 

45.44 

45.16 

44.88 

44.61 

44., 34 

44.07 

43.80 

43.54 

43.28 

43.a3 

42.77 

42.52 

42.28 

42.03 

41.79 

41.55 

41. 

41. 

40.85 

40.62 

40.40 


Sine. 


PPI"      Tans. 


8.718800 
721204 
72:3.595 
725972 
728:337 
730688 
7^3027 
7X5354 
737667 
739969 
742259 

8.744.536 
746802 
7490,55 
751297 
7,5a52S 
75574 
7579,55 
760151 
762337 
764511 

8.766675 
768828 
770970 
773101 
77.522:3 
777,333 
779434 
781.524 
7a%05 
78.567,5 

8.7877.36 
789787 
791828 
79:38.59 
795881 
797894 
799897 
801892 
80:3876 
80.Wj2 

8.807819 
809777 
811726 
81.3667 
81,5599 
817522 
8194:36 
8213^13 
82:3240 
82:130 

8.827011 
828884 
8:30749 
832G07 
8:344.56 
830297 
8381:30 
8399.56 
841774 
843585 


40.06 
39.84 
J9.62 
39.41 
39.19 
.38.98 
;38.77 
.38.57 
.38.36 
,38.16 
37.96 
:37.76 
;37.,56 
,37.37 
.37.17 
.36.98 
.36.80 
.36.61 
36.42 
,36.24 
.36.06 
a5.88 
a5.70 
35.53 
,3.5.  a5 
35.18 
.35.01 
a4.84 
:34.67 
,a4..51 

:34.:i5 

.34.18 

ai.02 

.a3.8G 
.'3:3.70 
:3;3.,54 
:i3.39 

:3;3.23 
:33.08 
.32.93 
.32.78 
32.63 
.32.49 
32.34 
32.19 
32.05 
:31.91 
31.77 
:31.63 
:31.49 
31.  a5 
31.22 
,31.08 
.30.95 
30.82 
30.69 
.30.56 
30.43 
,30.30 
30.17 


Cosinf 


PPI" 


PPl'     M 


8.719396 
721806 
724204 
726588 
728959 
731317 
7a3663 
735996 
738317 
740626 
742922 

8.74.5207 
747479 
749740 
751989 
754227 
756453 
7.58668 
760872 
7(5.3065 
765246 

8.767417 
709578 
771727 
77:3866 
775995 
778114 
780222 
782,320 
784408 
78(5486 

8.788.554 
790613 
792602 
794701 
79(5731 
798752 
800763 
8027(5.5 
8047.58 
80(5742 

8.808717 
810(]8.3 
812m  1 
814.589 
816529 
818461 
820:»1 
822298 
824205 
826103 

8.827992 
829874 
8:31748 
83361:? 
8a>471 
8.37321 
&39163 
840998 
842825 
844644 


40.17 
39.95 
39.74 
.39.52 
:39.30 
39.09 
:38.89 
38.68 
.38.48 
38.27 
:38.07 
37.87 
,37.68 
:37.49 
37.29 
37.10 
36.92 
36.73 
36..55 

.m 

.18 
:36.00 
.a5.83 
.a5.65 

.a5.48 

,a5.3i 
.a5.i4 

:34.J>7 
31.80 
34.64 
:34.47 
.34.;31 
:34.15 
.a3.99 

:a}.,s;3 
:5:3.68 
.a3..52 

:}3.37 

,3:3.22 

:33.07 

,32.92 

;32, 

:32.62 

;32.48 

:32.a3 

32.19 

:52.05 

,31.91 

.31.77! 

:31.6:3 

31.-50 

31.. 36 

31.23 

.31.10 

30.96 

:30.83 

.30.70 

30.57 

30.45 

30.32 


(^otang. 


PPI" 


378 


»«" 


1!_ 

"m. 
0 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 


85" 


TABLE   IV.— LOGARITHMIC 


50 


Sino 


8.843585 
845387 
847183 
848971 
8.50751 
852525 
854291 
856049 
857801 
85954G 
861283 

8.863014 
864738 
866455 
868165 


871565 
873255 
874938 
876615 
87828.5 
8.879949 
881607 
883258 
884903 
886542 
888174 


891421 
893035 
894643 
8.896246 
897842 


901017 
902.596 
904169 
905736 
907297 
908853 
910404 

8.911949 
913488 
915022 
916550 
918073 
919591 
921103 
922610 
924112 
92;5609 

8.927100 
928587 
930068 
931544 
983015 
934481 
935942 
937398 
9-38850 
940296 


Cosine. 


PPl' 


.30.05 

29.92 

29.80 

29.67 

29.-55 

29.43 

29.31 

29.19 

29.07 

28.96 

28.84 

28.73 

28.61 

28.50 

28. 

28.28 

28.17 

28.08 

27.95 

27.84 

27.73 

27.63 

27.52 

27.42 

27.31 

27.21 

27.11 

27.00 

26.90 

26.80 

26.70 

26.60 

26.51 

26.41 

26.31 

26.22 

26.12 

26.03 

25.93 

25.84 

25.75 

25.66 

25.56 

25.47 

25.38 

25.29 

25.20 

25.12 

25.03 

24.94 

24.86 

24.77 

24. 

24.60 

24.52 

24.43 

24.35 

24.27 

24.19 

24.11 


PPl' 


Tang.   PPl"  M 


8.844644 
846455 
848260 
850a57 
851846 


855403 
857171 


862433 

8.864173 

865906 

867632 


871064 
872770 
874469 
876162 
877849 
879529 
8.881202 


8845.30 
886185 
887833 
889476 
891112 
892742 
894366 


8.897596 
899203 
900803 
902398 


905570 
90714" 
908719 
910285 
911846 

8.913401 
914951 
91649.5 
918034 
919568 
921096 
922619 
924136 
925649 
927156 

8.928658 
9301-5;5 
931647 
93;31.34 
934616 
9:36093 
937-565 
939032 
940494 
941952 


Cotanar. 


30.19 

30.07 

29.95 

29.82 

29.70 

29., 58 

29.46 

29.  a5 

29.23 

29.11 

29.00 

28.88 

28.77 

28.66 

28.54 

28.43 

28.32 

28.21 

28.11 

28.00 

27.8 

27.7 

27.68 

27.58 

27.47 

27.37 

27.27 

27.17 

27.07 

26.97 

26.87 

26.77 

26.67 

26.58 

26.48 

26.38 

26.29 

26.20 

26.10 

26.01 

25.92 

25.83 

25.74 

25.65 

25.56 

25.47 

25.38 

25.30 

25.21 

25.12 

25.03 

24.95 

24.87 

24.78 

24.70 

24.62 

24.53 

24.45 

24.37 


PPl 


M. 


Sir 


8.940296 
941738 
943174 
944606 
946034 
947456 
948874 
950287 
951696 
953100 
954499 

8.95.5894 
957284 
958670 
960052 
961429 
962801 
964170 
965534 


968249 
8.969600 
97094' 
972289 
973628 
974962 
976293 
977619 
978941 
980259 
981573 
8.9828&:} 
984189 
985491 
986' 
988083 
989374 
990660 
991943 
993222 
994497 
8.995768 
997036 


999560 
9.000816 
002069 
.  003318 
004563 
005805 
007044 
9.008278 
009510 
010737 
011962 
013182 
014400 
01-5()13 
016824 
018031 
019235 


CJosine. 


PJ'l" 


24.03 

23.94 

23.87 

23.79 

23.71 

23.63 

23.55 

23.48 

23.40 

23.32 

23.25 

23.17 

23.10 

23.02 

22.95 

22.88 

22.80 

22.73 

22.66 

22. 

22.52 

22.45 

22.38 

22.31 

22.24 

22.1 

22.10 

22.03 

21.97 

21. 

21.83 

21.77 

21.70 

21.6:3 

21.57 

21.50 

21.44 

21.38 

21.31 

21.25 

21.19 

21.12 

21.06 

21.00 

20.94 

20.88 

20.82 

20.76 

20.70 

20.64 

20.58 

20.52 

20.46 

20.40 

20.:34 

20.29 

20.2:3 

20.1 

20.12 

20.06 


PPl' 


Tang. 


PI* 


8.941952 
943404 
944852 
946295 
947734 
949168 
950597 
952021 
953441 
954856 
956267 

8.957674 
959075 
960473 
961866 
963255 
964639 
966019 
967394 
968766 
970133 

8.971496 
972855 
974209 
975560 
976906 
978248 
979586 
980921 
982251 
983577 

8.984899 
986217 
987532 
988842 
990149 
991451 
992750 
994045 
995337 
996624 

8.997908 
999188 

9.000465 
001738 
003007 
004272 
005534 
006792 
008047 
009298 

9.010546 
011790 
013031 
014268 
015502 
016732 
017959 
01918:3 
020403 
021620 


24.21 

24.13 

24.05 

23.97 

23.90 

23.82 

23.74 

23.66 

23.60 

23.51 

23.44 

23.37 

23.29 

23.22 

23.14 

23.07 

23.00 

22.93 

22.86 

22.79 

22.71 

22.65 

22.57 

22.51 

22.44 

22.37 

22,30 

22.23 

22.17 

22.10 

22.04 

21.97 

21.91 

21.84 

21.78 

21.71 

21.65 

21.68 

21.52 

21.46 

21.40 

21.34 

21.27 

21.21 

21.15 

21.09 

21.03 

20. 

20.91 

20.85 

20.80 

20.74 

20.68 

20.62 

20.56 

20.51 

20.45 

20.40 

20.33 

20, 


("otang. 


I'PI' 


374 


84« 


6« 


SINES  AND  TANGENTS. 


1^0 


Siiu!. 


9.01923.5 
020435 
021G32 
022825 
02401G 
023203 
02(J386 
027o(>7 
028744 
029918 
031089 

9-0322.")7 
0:^3421 
034582 
03)741 
03(5896 
038048 
039197 
040342 
01148.') 
012625 

9.013762 
044895 
016026 
0171.54 
048279 
049400 
0.50519 
05ia3.5 
052749 
0538.59 

9.054966 
056071 
057172 
058271 
059337 
0:)0460 
061.551 
0626:39 
063724 
064806 

9.0()o885 
066962 
068036 
069107 
070176 
071242 
072306 
07:3366 
074124 
075480 

9.07()533 
077.583 
078631 
079676 
080719 
0^1759 
082797 
08:38:32 
084864 
O.S;5894 


Pl'l" 


20.00 
19.95 
19.89 
19.a4 
19.78 
19.73 
19.67 
19.62 
19.57 
19.  .51 
19.46 
19.41 
19.36 
19.30 
19.2.5 
19.20 
19.15 
19.10 
19.05 
18.99 
18.94 
18.89 
18.84 
18.80 
18.75 
18.70 
18.65 
18.60 
18.;55 
18.. 50 
18.45 
18.41 
18.:36 
18.31 
18.27 
18.22 
18.17 
18.13 
18.08 
18.04 
17.99 
17.94 
17.90 
17.86 
17.81 
17.77 
17.72 
17.68 
17.63 
17.. 59 
17.5.5 
17.50 
17.46 
17.42 
17.38 
17.33 
17.29 
17.2.5 
17.21 
17.17 


Cosine. 


Tan  J 


PPl" 


.021620 
0228;34 
024044 
02.52.51 
026455 
027655 
0288.52 
0:30046 
0.31237 
03242.5 
a3:3609 

.034791 
ft3.5969 
0:37144 
0a8316 
0394,S5 
040651 
041813 
042973 
0441.30 
045281 

.0464:34 
047.582 
048727 
049869 
051008 
0,52144 
05,3277 
054407 
0555:3.5 
0566.59 

.057781 
0.58900 
060016 
0611:30 
062240 
063348 
0644.53 
06.555(5 
0666.55 
0677.52 

1.068846 


PP 


071027 
0721131 
0731971 
0742781 
07.53.56 
07&132 
077.505 
078.576 
).079644 
080710 
081773 
082833 
083891 
084947 
086000 
087050 
088098 
089144 


20.23 
20.17 
20-11 
20.06 
20.00 
19.95 
19.90 
19.85 
19.79 
19.74 
19.69 
19.64 
19.58 
19.5:3 

19.48 

19.43 

19.38 

19.-33 

19.28 

19.23 

19.18 

19.13 

19.08 

19.03 

18.98 

18.93 

18.89 

18.84 

18.79 

18.74 

18.70 

18.6.5 

18.60 

18.55 

18.51 

18.46 

18.42 

18.:37 

18.33 

18.28 

18.24 

18.19 

18.15 

18,10 

18.06 

18.02 

17.97 

17, 

17.89 

17.84 

17.80 

17.76 

17.72 

17.67 

17.63 

17.59 

17.55 

17., 51 

17.47 

17.43 


Cotang.  I  PPl"  M 


I  PPl" 


9.085894 
086922 
087947 
088970 
089990 
091008 
092024 
093037 
094047 
09.5056 
096062 

9.097065 
098066 
099065 
100062 
1010^56 
102048 
1030:37 
104025 
105010 
10.5992 

9.106973 
1079.51 
108927 
109901 
110873 
111842 
112809 
113774 
114737 
11.5698 

9. 116(5.56 
117613 
118567 
119519 
120469 
121417 
122362 
123306 
124248 
125187 

9.126125 
127000 
127993 
128925 
129854 
130781 
131706 
132630 
133551 
1.34470 

9.13.5,387 
136303 
1,37216 
138128 
1,39037 
139944 
140850 
1417.54 
142655 
143555 


17.13 
17.09 
17.04 
17.00 
16.96 
16.92 
16.88 
16.84 
16.80 
16.76 
16.73 
16.68 
16.65 
16.61 
16.57 
16.53 
16.49 
16.45 
16.41 
16.38 
16.31 
16.30 
16.27 
16.2:3 
16.19 
16.16 
16.12 
16.0! 

16.  o; 

16.01 
15.97 
15.94 
15.90 
15.87 
15.83 
15.80 
15.76 
15.73 
15.  ()9 
15.66 
15.62 
15..59| 
15.56! 
15.. 52 
15.491 
15.451 
15.42! 
15.39 
15.35 
15.  .32 
15.29 
15.25 
15.22 
1.5.19 
15.16 
15.12 
15.09 
15.06 
15.03 
15.00 


'lai 


.089144 
090187 
091228 
092266 
093302 
094336 
095367 


097422 
098446 
099468 

).  100487 
101,504 
102519 
103532 
104542 
105550 
106556 
107559 
108,560 
109.559 

).110556 
111.551 
112.543 
113533 
114521 
115507 
116491 
117472 
118452 
119429 

9.120404 
121377 
122M8 
123317 
124284 
12.5249 
126211 
1271 
1281.30 
129087 

9.1:30041 
130994 
131944 
1:32893 
133839 
134784 
ia5726 
1136667 
13760.5 
1.38542 

9.13947 
140409 
14iai0 
142269 
143196 
144121 
145044 
145966 
146885 
147803 


PP 


Cotang. 


17.:38 
17.34 
17.30 
17.27 
17.22 
17.19 
17.15 
17.11 
17.07 
17.03 
16.99 
16.95 
16.91 
16.87 
16.84 
16.80 
16.76 
16.72 
16.69 
16.65 
16.61 
16.58 
16.54 
16.50 
16.46 
16.43 
16.39 
16.36 
16.:32 
16.29 
16.25 
16.22 
16.18 
16.15 
16.11 
16.07 
16.04 
16.01 
15.97 
1.5.94 
15.91 
15.87 
15.84 
15.81 
15.77 
15.74 
15.71 
15.67 
15.64 
15.61 
15.58 
15.55 
15.51 
15.48 
15.45 
15.42 
15.39 
15.  a5 
15.32 
15.29 


IM. 


PPl' 


83" 


375 


M. 


M. 
0 

1 


SI' 


TABLE   IV.— LOGARITHMIC 


9« 


iSinc.  PPl"  TauK.  PPl"  BL_ 
60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 


9.143555 
144453 
145349 
146243 
147136 
148026 
148915 
149802 
150686 
151569 
152451 

9.153330 
154208 
155083 
155957 
156830 
157700 
158569 
159435 
160301 
161164 

9.162025 
162885 
163743 
164600 
1&5454 
166307 
167159 
168008 
168856 
169702 

9.170547 
171389 
172230 
173070 
173908 
174744 
175578 
176411 
177242 
178072 

9.178900 
179728 
180r»l 
181374 
182196 
183016 
183834 
184651 
185466 
180280 

9.187092 
187903 
188712 
189519 
19032.5 
191130 
191933 
192734 
193534 
194332 


14.96 

14.93 

14.90 

14.87 

14.84 

14.81 

14.78 

14.75 

14.72 

14.69 

14.66 

14.63 

14.60 

14.57 

14.54 

14.51 

14.48 

14.45 

14.42 

14.39 

14.36 

14.33 

14.30 

14.27 

14.24 

14.22 

14.19 

14.16 

14.13 

14.10 

14.07 

14.05 

14.02 

13.99 

13.96 

13.94 

13.91 

13.88 

13.86 

13.83 

13.80 

13.77 

13.74 

13.72 

13.69 

13.06 

13.64 

13.61 

13.59 

13.56 

13.53 

13.51 

13.48 

13.46 

13.43 

13.41 

13.C 

13.^ 

13.^ 

13.30 


Cosine. 


PPV 


9.147803 
148718 
149632 
150544 
151454 
152363 
153269 
154174 
155077 
155978 
156877 

9.157775 
158671 
159565 
100457 
101347 
162236 
163123 
164008 
164892 
105774 

9.166654 
167532 
168409 
169284 
170157 
171029 
171899 
172767 
173634 
174499 

9.175362 
176224 
177084 
177942 
178799 
179655 
180508 
181360 
182211 
183059 

9.18390 
184752 
18559 
186439 
187280 
188120 
188958 
189794 
190629 
191462 

9.192294 
193124 
193953 
194780 
195606 
196430 
197253 
198074 
198894 
199713 


15.26 

15.23 

15.20 

15.17 

15.14 

15.11 

15.08 

15.05 

15.02 

14.99 

14.96 

14.93 

14.90 

14.87 

14.84 

14.81 

14.79 

14.76 

14.73 

14.70 

14.67 

14.64 

14.61 

14.58 

14.55 

14.53 

14.50 

14.4 

14.44 

14.42 

14. 

14. 

14.33 

14.31 

14.28 

14.25 

14.23 

14.20 

14.17 

14.15 

14.12 

14. 

14.07 

14.04 

14.02 

13.99 

13.96 

13.93 

13.91 

13.89 

13.86 

13.84 

13.81 

13.79 

13.76 

13.74 

13.71 

13.69 

13.66 

13.64 


('Otana. 


PPl' 


Sine. 


9.194332 
195129 
195925 
19071?) 
197511 
198302 
199091 
199879 
200066 
201451 
202234 

9.203017 
203797 
204577 
205354 
206131 
206906 
20767! 
208452 
209222 
209992 

9.210760 
211526 
212291 
213055 
213818 
214579 
215338 
216097 
216854 
217609 

9.218363 
219116 
219868 
220618 
221367 
222115 
222861 
223606 
224349 
225092 

9.225833 
226ij73 
227311 
228048 
228784 
229518 
230252 
2;30984 
231714 
232444 

9.233172 
233899 
2M625 
235^49 
236073 


PPl' 


237515 
238235 


239670 


13.28 

13.26 

13.23 

13.21 

13.18 

13.16 

13.13 

13.11 

13.08 

13.06 

13.04 

13.01 

12.99 

12.96 

12.94 

12.92 

12.89 

12.87 

12. 

12.82 

12.80 

12.78 

12.75 

12.73 

12, 

12.68 

12.66 

12.64 

12.62 

12.59 

12.57 

12.55 

12.53 

12.50 

12.48 

12.46 

12.44 

12.42 

12.39 

12.37 

12.35 

12.33 

12.81 

12.28 

12.26 

12.24 

12.22 

12.20 

12.18 

12.16 

12.14 

12.12 

12.09 

12.07 

12.05 

12.03 

12.01 

11.99 

11.97 

11.95 


Tail-.   PPl"  JNf. 


9.199713 
200529 
201345 
202159 
202971 
203782 
204592 
2a5400 
200207 
207013 
207817 

9.208619 
209420 
210220 
211018 
211815 
212611 
213405 
214198 
214989 
215780 

9.210568 
217356 
218142 
218926 
219710 
220492 
221272 
222052 
222830 
223607 

9.224382 
225156 
22.5929 
226700 
227471 
228239 
229007 
229773 
230539 
231.302 

9.232065 
232826 
233586 
234345 
2,35103 
235859 
236614 
237368 
238120 
238872 

9.2^9622 
240371 
241118 
241865 
242610 
243354 
244097 
244839 
245579 
246319 


13.61 
13.59 
13.57 
13.54 
13.52 
13.49 
13.47 
13.45 
13.42 
13.40 
13.38 
13.  a5 
13.33 
13.31 
13.28 
13.26 
13.24 
13.21 
13.19 
13.17 
13.15 
13.12 
13.10 
13.08 
13.06 
13.03 
13,01 
12.99 
12.97 
12.94 
12.92 
12.90 
12.88 
12.86 
12.  &4 
12.81 
12.79 
12.77 
12.75 
12.73 
12.71 
12.69 
12.67 
12.65 
12.62 
12.60 
12.58 
12.56 
12.54 
12.52 
12.50 
12.48 
12.46 
12.44 
12.42 
12.40 
12.38 
12.36 
12.34 
12.32 


Cosine.  I'Pl"  Cotang.  PPl"  M. 


376 


80° 


io< 


SINES  AND  TANGENTS. 


11 « 


30 
31 
32 
33 
31 

a3 

m 

37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.239670 
240386 
241101 
211814 
212.526 
2mS7 
213947 
214656 
215:363 
2K30)9 
216775 

9.217478 
218181 
2188.^3 
249583 
250282 
2509.S0 
251677 
252373 
253087 
2.5:3761 

9. 214453 
255144 
2.>58:31 
256.52-3 
257211 
2-57898 
258;5a3 
259268 
259951 
2606.33 

9.261314 
261994 
262673 
26;i3.51 
231027 
264703 
265377 
266a51 
286723 
287395 

9.26806.' 
268734 
269102 
27003! 
2707*5 
271400 
272011 
272726 
273;3S8 
274019 

9.274708 
27.5337 
276025 
276681 
277,337 
277991 
278645 
279297 
279948 
280599 


I'I'l' 


11.93 
11.91 
11.89 
11.87 
11.85 
11.83 
11.81 
11.79 
11.77 
11.75 
11.73 
11.71 
11.69 
11.67 
11.65 
11.63 
11.61 
11.59 
11..58 
11.56 
11.54 
ll.,52 
11.50 
11.48 
11.46 
11.44 
11.42 
11.41 
11.39 
11.37 
11.  a5 
11.33 
11.31 
11.30 
11.28 
11.26 
11.24 
11.22 
11.20 
11.19 
11.17 
11.15 
11.13 
11.12 
11.10 
11.08 
11.06 
11.05 
11.03 
11.01 
10.99 
10.98 
10.96 
10.94 
10.92 
10.91 
10.89 
10.87 
10.86 
10.84 


Tuner. 


9.246319 
247a57 
247794 
248530 
249264 
249998 
250730 
2.51461 
2.52191 
2.52920 
253648 

9.2.51.374 
2-55100 
255824 
2.56517 
2)7269 
257990 
258710 
259429 
260146 
26086:3 

9.2(31578 
262292 
26.30a5 
263717 
264428 
265138 
265847 
2665.55 
267261 
267967 

9.268671 
269375 
270077 
270779 
271479 
272178 
272876 
273573 
274269 
274964 

9.275658 
276351 
277043 
277734 
27&121 
279113 
279801 
280488 

■  281174 
281858 

9.282542 
283225 
28390' 
284588 
28.5268 
285947 
286624 
287301 
287977 
288652 


PP 


12.30 

12.28 

12-26 

12.24 

12.22 

12.20 

12.18 

12.17 

12.15 

12.13 

12.11 

12.09 

12.07 

12.0.5 

12.03 

12,01 

12.00 

11.98 

11.96 

11.94 

11.92 

11.90 

11.89 

11.87 

11.85 

11.83 

11.81 

11.79 

11.78 

11.76 

11.74 

11.72 

11.70 

11.69 

11.67 

11.65 

11.64 

11.62 

11. 

11.58 

11.57 

11..55 

11.53 

11.51 

11.50 

11.48 

11.47 

11.45 

11.43 

11.41 

11.40 

11.38 

11.36 

11.35 

11.33 

11 

11.30 

11.28 

11.26 

11.25 


M. 


M. 


SiiK 


9.280599 
281218 
281897 
282544 
283190 


2»4480 
285124 
28.5766 
286408 
287048 


2889&1 
289600 
290236 
290870 
291504 
292137 
292768 


Cosine.    PPl"    Cotang.    PPl"    M.      M.      Cosine.     PPl"    Cotang.    PPl"    M. 


9.294029 
294658 
295286 
2ft5913 
296,539 
2971frl 
297788 
298412 
2990ai 
299<}55 

9.300276 
300895 
301514 
302132 
302748 
3033ty 
303979 
301593 
305207 
305819 

9.306430 
307041 
307650 
308259 
308867 
309474 
310080 
310685 
311289 
311893 

9.312495 
31:3097 
313698 
314297 
314897 
315195 
316092 
316689 
317284 
317879 


PPl" 


10.82 
10.81 
10.79 
10.77 
10.76 
10.74 
10.72 
10.71 
10.69 
10.67 
10.66 
10.64 
10.63 
10.61 
10.59 
10.58 
10.,56 
10.54 
10.. 53 
10.51 
10.. 50 
10.48 
10.46 
10.45 
10.43 
10.42 
10.40 
10.39 
10.37 
10.36 

10.  ai 

10.-32 
10.-31 
10.29 
10.28 
10. 2« 
10.25 
10.23 
10.22 
10.20 
10.19 
10.17 
10.16 
10.14 
10.13 
10.11 
10.10 
10.08 
10.07 
10.06 
10.04 
10.03 
10.01 
10.00 
9.98 
9.97 
9.96 
9.04 
9.93 
9.91 


.288652 
289326 


290671 
291342 
292013 
292682 
293350 
294017 
2946H4 
295349 

.296013 
296677 
297339 
298001 
298662 
299322 
2fi9980 
300638 
301295 
301951 

.302607 
30.3261 
303914 
301.567 
305218 
805869 
306519 
30716K 
307816 
30846:3 

.5309109 
309751 
310399 
311012 
311685 
312327 
312968 
313608 
31424 
314885 

.315,523 
316159 
316795 
317430 
318064 
318697 
319330 
319961 
320592 
321222 

.321851 
322479 
323106 


324358 


32560: 
326231 
326853 
327475 


PPl' 


11.23 

11.22 

11.20 

11.18 

11.17 

11.15 

11.14 

11.12 

11.11 

11.09 

11.07 

11.06 

11.01 

11.03 

11.01 

11.00 

10.98 

10.96 

10.95 

10.93 

10.92 

10.90 

10.89 

10.87 

10.86 

lO.gl 

10.83 

10.81 

10.80 

10.78 

10.77 

10.75 

10.74 

10.73 

10.71 

10.70 

10.68 

10.67 

10.65 

10.64 

10.62 

10.61 

10.60 

10.58 

10.5' 

10.55 

10.51 

10.53 

10.. 51 

10.50 

10.48 

10.47 

10.45 

10.44 

10.43 

10.41 

10.40 

10 

10.37 

10.36 


79' 


Trio:.— 32. 


377 


7H" 


1*J0 


TABLE   IV.— LOGARITHMIC 


13* 


Sine. 


.817879 
318473 
319066 
319658 
320249 
320840 
321430 
322019 
5226071 
323194 
323780 

.324366 
324950 
325534 
326117 
326700 
327281 


328442 
329021 
329599 

.330176 
330753 
331329 
331903 
332478 
333051 
333624 
334195 
334767 
335387 

.335906 
336475 
337043 
337610 
338176 
338742 
389307 
389871 
340434 
340996 

.341558 
^42119 
342679 
343239 
343797 
344355 
344912 
345469 
346024 
34&579 

.347134 
347687 
348240 
348792 
349343 


350443 
350992 
351540 

352088 


Cosine.     PPl 


9.87 
9.86 
9.84 
9.83 
9.82 
9.80 
.79 
9.77 
9.76 
9.75 
9.73 
9.72 
9.70 
9.69 
9.68 
9.66 
9.65 
9.64 
9.62 
9.61 
9.60 
9.58 
9.57 
9.56 
9.54 
9.53 
9.52 
9.50 
9.49 
9.48 
9.46 
9.45 
9.44 
9.43 
9.41 
9.40 
9.89 
9.87 
9.36 
9.85 
9.34 
9.82 
9.31 
9.30 
9.29 
9.27 
9.26 
9.25 
9.24 
9.22 
9.21 
9.20 
9.19 
9.17 
9.16 
9.15 
9.14 
9.13 


rang.     PPl"    M 


9.327475 
328095 
328715 
329884 
329953 
330570 
3:51187 
3318a3 
332418 
33;?083 
3&8646 

9.384259 
834871 
335482 
336098 
386702 
387311 
387919 
338527 
339138 
a39739 

9.340344 
340948 
341552 
342155 
342757 
348858 
348958 
344;558 
345157 
345755 

9.846358 
346949 
347.545 
348141 
348735 
349829 
349922 
a50514 
351106 
351097 

9.352287 
a32876 
353465 
354053 
354640 
355227 
a55813 
356398 


357566 
9.358149 
a">8731 
a-^9318 
3.39898 
360474 

miorjs 

361632 
362210 
362787 
363364 


10.85 

10.33 

10.32 

10.80 

10.29 

10.28 

10.26 

10.25 

10.24 

10.28 

10.21 

10.20 

10.19 

10.17 

10.16 

10.15 

10.13 

10.12 

10.11 

10.10 

10.08 

10.07 

10.06 

10.04 

10.08 

10.02 

10.00 

9.99 

9.98 

9.97 

9.96 

9.94 

9.93 

9.92 

9.91 

9.90 

9.88 

9.87 

9.86 

9.85 

9.88 

9.82 

9.81 

9.80 

9.79 


.75 


9. 

9. 

9.73 

9.71 

9.70 

9.69 

9.68 

9.67 

9.66 

9.65 

9.63 

9.62 

9.61 


Cotang.    PPl"    M 


M, 


9.332088 
352685 
a53181 
a33720 
a34271 
354815 

a55a38 

355901 
356448 
356984 
357524 

9.a38064 
a38603 
359141 
359678 
360215 
360752 
361287 
361822 
362356 
362889 

9.368422 
368954 
36448.3 
365016 
365546 
366075 
366604 
367181 
3676.59 
368185 

9.368711 
809236 
369761 
370285 
370808 
371380 
371852 
372378 
372894 
373414 

9.373988 
374452 
374970 
375487 
370008 
376519 
377035 
377549 
378063 
378577 

9.379089 
379601 
380113 
380624 
381134 
381643 
382152 
382661 
383168 
383675 


PPl' 


M.  Cosine.  PPl" 


9.11 
9.10 
9.09 
9.08 
9.07 
9.a3 
9.04 
9.08 
9.02 
9.01 
8.99 
8.98 
8.97 
8.96 
8.95 
8.98 
8.92 
8.91 
8.90 
8.89 
8.88 
8.87 
8.85 
8.84 
8.88 
8.82 
8.81 
8.80 
8.79 
8.77 
8.76 
8.75 
8.74 
8.73 
8.72 
8.71 
8.70 
8.69 
8.67 
8.66 
8.65 
8.64 
8.03 
8.62 
8.61 
8.60 
8.59 
8.58 
8.57 
8.56 
8.54 
8.53 
8.52 
8.51 
8.50 
8.49 
8.48 
8.47 
8.46 
8.45 


Tans. 


.868364 
368940 
364515 
365090 
365664 
366237 
366810 
367382 
367953 
36&524 
369094 

.369668 
370282 
370799 
871367 
371938 
372499 
3730(M 
373629 
374193 
374756 

.375319 
375881 
376442 
377003 
377563 
378122 
378 
379239 
379797 
380854 

.380910 
381466 
382020 
382575 
383129 
383082 
384234 
384786 
385337 
385888 

.386438 
386987 
387586 
388084 
388031 
389178 
889724 
390270 
390815 
391360 

.391903 
392447 
392989 
39a>]l 
394078 
394614 
3951.54 
395694 
396233 
396771 


9.00 
9.69 
9.58 
9.57 
9.55 
9.54 
9.53 
9.52 
9.51 
9.50 


9.46 
9.45 
9.44 
9.43 
9.42 
9.41 
9.40 
9.39 
9.38 
9.37 
9.35 
9.34 
9.38 
9.32 
9.31 
9.30 
9.29 
9.28 
9.27 
9.26 
9.25 
9.24 
9.23 
9.22 
9.21 
9.20 
9.19 
9.18 
9.17 
9.15 
9.14 
9.18 
9.12 
9.11 
9.10 
9.09 
9.08 
9.07 
9.06 
9.ft5 
9.04 
9.03 
9.02 
9.01 
9.00 
8.99 
8.98 
8.97 


Cotang.    PPl"    M 


7^' 


378 


•76^ 


140 


SINES  AND  TANGENTS. 


150 


31. 


8ii 


0  9. 

1 

2 

3 

4 

5 

6 

7 


399575 
400062 
400549 
4010a5 
401520 
402005 
402489 
402972 
4034.55 
.403938 
404420 
404901 
40.5382 
405802 
406341 
406820 
407299 
407777 
408254 
.408731 
409207 
409682 
410157 
4106.32 
411106 
411.579 
4120.52 
412.521 
412996 


8.43 
8.42 
8.41 
8.40 


.383675  L  ,. 

3841821""'^ 

384687 

385192 

385697 

386201 

386704 

387207 

387709 

3.S8210 

388711 

389211 

389711 

390210 

390708 

391206 

391703 

392199 

392695 

393191 

39;}685 

394179 

394673 

395166 

395658 

3961.50 

39<3641 

397132 

397621 

398111 

398600 


8.37 
8.36 
8.35 
8.34 
8M 
8.32 
8.31 
8.30 
8.28 
8.27 
8.26 
8.25 
8.24 
8.2;^ 
8.22 
8.21 
8.20 
8.20 
8.18 
8.17 
8.17 
8.16 
8.15 
8.14 
8.13 
8.12 
8.11 
8.10 
8.09 
8.08 
8.07 
8.08 
8.05 
8.04 
8.03 
8.02 
8.01 
8.00 
7.99 
7.98 
7.97 
7.96 
7.95 
7.94 
7.94 
7.93 
7.92 
7.91 
7.90 
7.89 
7.88 
7.87 
7.80 


PPl' 


9.39.')771 


17846 


398919 
3994.55 


40a524 
4010.58 
401591 
402124 

9.402ft50 
403187 
403718 
404249 
404778 
40.5:^08 
4ft58;36 
406364 
406892 
407419 

9.407945 
408471 


409.521 
410045 
4ia569 
411092 
411615 
412137 
412658 

9.413179 
413699 
414219 
414738 
4152.57 
415775 
416293 
416810 
417326 
417842 

9.4183.58 
418873 
419387 
419901 
420415 
420927 
421440 
421952 
422463 
422974 

9.423484 
423993 
424.503 
425011 
425519 
426027 
426;534 
427041 
427547 
428052 


PPl"  M 


8.96 
8.95 
8.94 
8.93 
8.92 
8.91 
8.90 
8.89 
8.8.S 
8.87 
8.86 
8.a5 
8.84 
8.83 
8.82 
8.81 
8.80 
8.79 
8.78 
8.77 
8.76 
8.75 
8.74 
8.74 
8.73 
8.72 
8.71 
8.70 
8.69 
8.68 
8.67 
8.66 
8.6.5 
8.64 
8.64 
8.63 
8.62 
8.61 
8.60 
8.59 
8..58 
8.-57 
8.  ,56 
8.55 
8.55 
8..54 
8.53 
8.52 
8.51 
8.,50 
8.49 
8.48 
8.48 
8.47 
8.46 
8.45 
8.44 
8.43 
8.43 


M. 


Sine.  I  PPl" 


9. 


412996 
413467 
413938 
414408 
414878 
41.5347 
415815 
416283 
416751 
417217 
4176»4 
418150 
418615 
419079 
4ia>44 
420007 
420470 


421395 
421857 
422318 

).  422778 
423238 
423697 
424156 
424615 
42507 
425530 
425987 
426443 
426899 

).  427354 
427809 
428263 
428717 
429170 
429623 
430075 
430527 
430978 
431429 

5.431879 
432329 
432778 
433226 
433675 
4^^122 
434569 
4;i3016 
43.5462 
4a5908 

).  436353 
436798 
437242 
437686 
438129 
438572 
439014 
439456 
439897 
440338 


^50 


rotang.  PPl"  m.      M.   Cosi 

379 


7.85 
7.84 
7.83 
7.83 
7.82 
7.81 
7.80 
7.79 
7.78 
7.77 
7.76 
7.75 
7.74 
7.73 
7.73 
7.72 
7.71 
7.70 


7.67 
7.67 
7.66 
7.65 
7.64 
7.63 
7.62 
7.61 
7.60 
7.60 
7.59 
7.58 
7.57 
7.56 
7.55 
7.54 
7.53 
7.53 
7.52 
7.51 
7.50 
7.50 
7.49 
7.48 
7.47 
7.46 
7.45 
7.45 
7.44 
7.43 
7.42 
7.41 
7.40 
7.40 
7.39 
7.38 
7.37 
7.. 36 
7.35 
7.35 


Tang. 


).428a52 
428.558 
429062 
429566 
430070 
4a573 
431075 
431577 
432079 
432580 


.43a580 
434080 
434579 
435078 
435576 
436073 
436,570 
4370()7 
437.56;j 


9.438554 
439048 
43a543 
44003*) 
440529 
441022 
441514 
442006 
442497 
442988 

9.443479 
443968 
444458 
444947 
44^35 
445923 
446411 


447384 
447870 

9.448356 
448841 
449326 
449810 
450294 
450777 
451260 
451743 
452225 
452706 

9.453187 
453668 
454148 
454628 
455107 
455586 
456064 
456542 
457019 
457496 


PPl"  Cotaiip.  PPl"  M. 


-540 


16" 


TABLE   IV.— LOGARITHMIC 


17° 


Sine. 


9.440338 
440778 
441218 
44ia58 
442096 
4425*5 
442973 
443410 
44:3847 
444284 
444720 

9.44515.5 
44.5590 
446025 
4464.59 


447326 
447759 
448191 
448623 
4490.54 

9.4494a5 
449915 
450345 
4,50775 
451204 
451632 
4.52060 
452488 
452915 
453342 

9.453768 
454194 
454619 
4.55014 
455469 
455893 
4.56316 
456739 
457162 
457584 

9.458006 
458427 
458848 
459268 
4.59688 
460108 
460527 
460946 
461364 
461782 

9.462199 
462616 
46.3032 
463448 
463864 
464279 
464694 
465108 
465522 
465935 


PPi" 


7.34 

7.  as 

7.32 
7.31 
7.31 
7.30 
7.29 
7.28 
7.27 
7.27 

26 
7.25 
7.24 

23 
7.23 
7.22 
7.21 
7.20 
7.20 
7.19 
7.18 
7.17 
7.17 
7.16 
7.15 
7.14 
7.13 
7.13 
7.12 
7.11 
7.10 
7.10 
7.09 
7.08 
7.07 
7.07 
7.08 
7.05 
7.05 
7.04 
7.03 
7.02 
7.01 
7.00 
7.00 
6.99 
6.98 
6.98 
6.97 
6.96 
6.95 
6.95 
6-94 
6.93 
6.93 
6.92 
6.91 
6.90 


Cosino.     PPI 


Tans.     IPPl 


9.4.57496 
457973 
458449 
45892.5 
459400 
459875 
460349 
460823 
461297 
461770 
462242 

9.462715 
463186 
463&58 
464128 
464599 
4&5069 
46.>539 
466008 
466477 


9.467413 

467880 
468347 
468814 


469746 
470211 
470676 
471141 
471605 

9.472069 
472532 
472995 
473457 
473919 
474381 
474842 
475303 
>175763 
476223 

9.476683 
477142 
477601 
478059 
478517 
478975 
479432 
479889 
480345 
480801 

9.481257 
481712 
482167 
482021 
483075 
48a529 
483982 
484435 
484887 
485339 


7.94 
7.93 
7.93 
7.92 
7.91 
7.90 
7.90 
7.89 
7.88 
7.88 
7.87 
7.86 
7.85 
7.8.5 
7.84 
7.83 
7.83 
7.82 
7.81 
7.80 
7.80 
7.79 
7.78 
7.78 
7.77 
7.76 
7.75 
7.75 
7.74 
7.73 
7.73 
7.72 
7.71 
7.70 
7.70 
7.70 
7.69 
7.68 
7.67 
7.67 
7.66 
7.65 
7.65 
7.64 
7.63 
7.63 
7.02 
7.61 
7.60 
7.60 
7.60 
7.59 
7.58 
7.57 
7.57 
7.56 
7.55 
7.55 
7.54 
7.58 


M.   M. 


60 


Cotans.  PPI"  M 


iMiie. 


I'Fl' 


Tang.   PPI"  M. 


466348 
466761 
467173 
467585 
467996 
468407 
468817 
469227 
469637 
470046 

9.470455 
470863 
471271 
471679 
472086 
472492 
472898 
473304 
473710 
474115 

9.474519 
474923 
475327 
475730 
476133 
476536 
476938 
477340 
477741 
478142 

9.478542 
478942 
479342 
479741 
480140 
480539 
480937 
481334 
481731 
482128 

9.482525 
482921 
483316 
483712 
484107 
484501 
484895 
485289 
485682 
486075 

9.486467 
486860 
487251 
487643 
488034 
488424 
488814 
489201 


6.88 
6.88 
6.87 
6.86 
6.85 
6.85 
6.84 
6.83 
6.83 
6.82 
6.81 
6.80 
6.80 
6.79 
6.78 
6.78 
6.77 
6.77 
6.76 
6.75 
6.74 
6.74 
6.73 
6.72 
6.72 
6.71 
6.70 
6.70 
6.69 
6.68 
6.67 
6.67 
6.66 
6.65 
6.65 
6.64 
6.63 
6.63 
6.62 
6.62 
6.61 
6.60 
6.59 
6.59 
6.58 
6.57 
6.57 
6.56 
6.55 
6.55 
6.54 
6.53 
6.53 
6.52 
6.51 
6.50 
6.50 
6.50 
6.49 
6.48 


485791 
486242 
486693 
487143 
487593 
488043 
488492 
488941 
489390 
489&S8 

9.490286 
490733 
491180 
491627 
492073 
492.519 
492965 
493410 
493854 
494299 

9.494743 
495186 
495630 
496073 
496515 


497399 
497841 
498282 
498722 

9.499163 
499603 
500042 
500481 
500920 
501359 
601797 
502235 
502672 
503109 

9.503546 
603982 
504418 
501854 
605289 
605724 
606159 
606593 
607027 
507460 

9.5078a3 
508326 
608759 
509191 
609622 
610064 
510485 
610916 
611346 
511776 


7.53 
7.52 
7.51 
7.51 
7.50 
7.49 
7.49 
7.48 
7.47 
7.47 
7.46 
7.46 
7,45 
7.44 
7.44 
7.43 
7.43 
7.42 
7.41 
7.40 
7.40 
7.40 
7.39 
7.38 
7.37 
7.37 
7.36 


60 


7.35 
7.34 
7.34 
7.33 
7.33 
7.32 
7.31 
7.31 
7.30 
7.30 
7.29 
7.28 
7.28 
7.27 
7.27 
7.26 
7.25 
7.25 
7.24 
7.24 
7.23 
7.22 
7.22 
7.22 
7.21 
7.20 
7.19 
7.19 
7.18 
7.18 
7.17 
7.17 


tS^ 


880 


Cosino.     PPI"    Cotang.    PPI"    M. 


IS' 


SINES  AND  TANGENTS. 


19^ 


Sill. 


.489982 
490:?71 
490759 
491147 
4915^5 
491922 
492308 


493081 
493466 
4938.51 
.4942-36 
494621 
495005 
49.5388 
49.5772 
4961.>4 
496.5:^7 


497:301 
497682 

9.498064 
498444 
498825 
499204 
499584 
4999^3 
500342 
500721 
501099 
501476 

9.. 501851 
502231 
502607 
502984 
50:«()0 
503735 
504110 
504485 
504860 
50.52;i4 

9.. 50.5608 
505981 
506354 
506727 
5070[)9 
507471 
507843 
508214 
50a585 
508956 

9.509326 
509696 
510065 
510134 
510803 
511172 
511540 
511907 
512275 
512&42 


M.   Cosine.  PPl 


6.48 

6.47 

.47 

.46 

.45 

.45 

.44 

6.43 

.43 

.42 

6.42 

6.41 

.40 

.40 


.38 
6.38 
6.37 
6.37 
6.36 
6.36 
6.:i5 

.34 
6.33 
6.33 
6.32 

.32 
6.31 
6.30 
6.:30 
6.29 
6.28 
6.28 
6.28 
6.27 
6.26 
6.26 
6.2.5 

.2.5 

.24 
6.23 
6.22 

.22 
6.22 
6.20 
6.20 
6.20 
6.19 
6.18 
6.18 
6.17 
6.17 
6.16 
6.15 
6.15 
6.15 
6.14 
6.13 
6.13 
6.12 


9.511776 
512206 
512635 
513064 
513493 
513921 
514349 
514777 
515204 
51.5631 
516057 

9.516484 
516910 
517;S3.5 
517761 
518186 
518610 
5190.34 
5194.58 
519882 
520.305 

9.520728 
521151 
521.573 
52199.5 
522417 
522838 
52:5259 
52.3680 
524100 
524.520 

9.524910 
5253.59 
52.5778 
526197 
526615 
5270:33 
527451 
527868 
52828.5 
528702 

9.529119 
529.535 
529951 
530366 
530781 
531196 
531611 
53202-5 
532439 
532853 

9.533266 
53:3679 
5:34092 
5^4504 
5:^916 
5a5:328 
535739 
536150 
536.561 
5369' 


PPl' 


.16 
7.16 
7.15 
7.14 
7.14 
7.13 
7.13 
7.12 
7.12 
7.11 
7.10 
7.10 
7.09 
7.09 
7.08 
7.08 
7.07 
7.06 
7.06 
7.a5 

.0.5 
7.04 
7.03 
7.03 
7.03 
7.02 
7.02 
7.01 
7.01 
7.00 
6.99 

.99 


6.97 
6.97 
6.96 
6.96 
.95 
6.95 
6.94 
6.93 
6.93 
6.93 
6.92 
6.91 
6.91 
6.90 
6.90 


6.88 
6.87 
6.87 
6.86 
6.86 
6.85 
6.85 
6.84 


M.   M 


9.512642 
513009 
513375 
513741 
514107 
514472 
514837 
515202 
51.5566 
515930 
51629-4 

9.516657 
517020 
517382 
517745 
518107 
518468 
5iaS29 
519190 
519.551 
519911 

9.520271 
520<i;31 
520990 
621349 
521707 
522066 
52*2424 
522781 
523138 
523495 

9.523852 
524208 
524;564 
524920 
525275 
525(530 
52.5984 


526693 
527046 

9.527400 
52775:3 
528105 
6284,58 
528810 
629161 
629513 
529864 
530215 
5305a" 

9.530915 
531265 
631614 
631963 
532312 
532661 
53:3009 
533357 
533704 
5^4052 


I'Pl' 


6.12 
6.11 
6.10 
6.10 
6.09 
6.08 
6.08 
6.07 
6.07 
6.06 
6.05 
6.05 
6.04 
6.04 
6.03 
6.02 
6.03 
6.02 
6.01 
6.00 
6.00 
6.00 
5.99 
5.98 
5.97 
5.98 
5.97 
5.96 
.5.95 
.5.95 
5.95 
5.94 
5.93 
5.93 
.5.92 
5.92 
5.90 
.5.91 
5.90 
5.89 
.5.90 
5.89 
5.87 
5.88 
.5.87 
.5.8.5 
5.86 
.5.85 
5.85 
5.84 
5.83 
5.83 
5.82 
.5.82 
5.82 
5.81 
5.80 
5.80 
5.80 
5.79 


ir 


Cotang.  I  PPl"    M.      M.      Cosino.     Pt 

381 


9.536972 
5:37382 
537792 
638202 
638611 
639020 
53W29 
5398^ 
540245 
640653 
641061 

9.5^41468 
541875 
542281 
542688 
643094 
64:3495) 
643905 
644310 
^4715 
645119 

9.645524 
645928 
646331 
546735 
547138 
647.S40 
547943 
548345 
648747 
649149 

9.549.550 
549951 
650352 
650752 
551153 
551552 
551952 
552351 
552750 
553149 

9.553548 
553946 
564344 
654741 
665139 
555536 
65593:3 
566329 
666725 
667121 

9.557617 
567913 
658308 
558703 
559097 
559491 
659885 
660279 
560673 
561066 


PPl"  M 


Cotang.  PPl"  M 


*jOo 


20» 


TABLE   IV.— LOGARITHMIC 


•210 


Sine. 


9.5S4052 
534399 
534745 
535092 
535438 
535783 
53(5129 
536474 
536818 
537163 
537507 

9.537851 
538194 
538538 


539223 
539565 
539907 
540249 
540590 
540931 

9.541272 
541613 
541953 
542293 
542632 
542971 
543310 
543649 
543987 
544325 

9.544663 
545000 
515338 
545674 
546011 
546347 
546683 
547019 
547351 
547689 

9.548024 
548359 
548693 
549027 
549360 
549693 
550026 
550359 
550692 
551024 

9.551356 
551687 
552018 
552349 
552680 
553010 
553341 
553670 
554000 
554329 


Cosine. 


PP 


5.78 
5.77 
5.77 
5.77 
5.76 
5.76 
5.75 
5.74 
5.74 
5.73 
5.73 
5.72 
5.72 
5.71 
5.71 
5.70 
5.70 
5.70 
5.69 
5.68 
5.68 
5.67 
5.67 
5.66 
5.65 
5.65 
5.65 
5.65 
5.64 
5.63 
5.63 
5.62 
5.62 
5.61 
5.61 
5.60 
5.60 
5.60 
5.59 
5.58 
5.58 
5.57 
5.57 
5.56 
5.55 
5.55 
5.55 
5.55 
5.54 
5.-53 
5.53 
5.52 
5.52 
5.52 
5.51 
5.50 
5.50 
5.50 
5.50 


PPl' 


Tang. 


9.561066 
5614,59 
561851 
562244 
562636 
563028 
563419 
563811 
564202 
564593 
564983 

9.565373 
565763 
56615;^ 
566542 
566932 
567320 
567709 


568486 
568873 

9.569261 
509648 
570035 
570422 
570809 
571195 
571581 
571967 
572352 
572738 

9.573123 
573507 
573892 
574276 
574660 
575044 
575427 
575810 
576193 
576576 

9.576959 
577341 
577723 
578104 
578486 
578867 
579248 
579629 
580009 


.580769 
581149 
581528 
581907 


582665 
583044 
583422 
583800 
584177 


Cotang. 


PPl" 


6.55 
6.54 
6.54 
6.53 
6.53 
6..53 
6.52 
6.52 
6.51 
6.51 
6.. 50 
6.50 
6.49 
6.49 
6.49 
6.48 
6.48 
6.47 
6.47 
6.46 
6.46 
6.45 
6.45 
6.45 
6.44 
6.44 
6.43 
6.43 
6.42 
6.42 
6.42 
6.41 
6.41 
6.40 
6.40 
6.39 
6.39 
6.39 
6.38 
6.38 
6.37 
6.37 
6.36 
6.36 

6.35 
6,35 
6.34 
6.34 
6.34 
6.33 
6.33 
6.32 
6.32 
6.32 
6.31 
6.31 
6.30 
6.30 


M. 


Sine.   PPl 


9.554329 
554658 
554987 
555315 
555643 
555971 
556299 
556626 
556953 
557280 
5.57606 

9.557932 
558258 
558583 
558909 
559234 
559558 
559883 
560207 
560531 
560855 

9.561178 
561501 
561824 
562146 
562468 
502790 
563112 
563433 
563755 
564075 

9.564396 
564716 
565036 
565356 
565676 
565995 
566314 
566632 
566951 
567269 

9.567587 
567904 
568222 
568539 
568856 
569172 
569488 
569804 
570120 
5704aj 

9.570751 
571066 
571380 
571695 
572009 
672323 
572636 
572950 
573263 
673575 


Cosine. 


5.48 
5.48 
5.47 
5.47 
5.47 
5.46 
5.45 
5.45 
5.45 
5.44 
5.43 
5.43 
5.42 
5.42 
5.42 
5.41 
5.41 
5.40 
5.40 
5.40 
5.39 
5.38 
5.38 
5.37 
5.37 
5.37 
5.36 
5.35 
5.35 
5.35 
5.34 
5.34 
5.33 
5.33 
5.33 
5.32 
5.32 
5.31 
5.31 
5.30 
5.30 
5.29 
5.29 
5.28 
5.28 
5.27 
5.27 
5.27 
5.26 
5.25 
5.25 
5.25 
5.24 
5.24 
5.23 
5.23 
5.23 
5.22 
5.21 
5.20 


PPl" 


Tang.   PPl"  JM. 


.5^177 
684555 
584932 
585309 
585686 
586062 
686439 
686815 
687190 
587566 
687941 

.588316 
588691 
689066 
589440 
589814 
59018S 
690562 
69093.5 
691308 
591681 

.592054 
692426 
592799 
693171 
593542 
593914 
594285 
594656 
596027 
695398 

.595768 
596138 
596508 
596878 
597247 
597616 
597985 
598354 
598722 
699091 

.699459 
599827 
600194 
600562 
600929 
601296 
601663 
602029 
602395 
602761 

.603127 
603493 
603858 
604223 
C04588 
604953 
605317 
605682 
606046 
606410 


Cotaiig.  PPl"  M 


69^ 


382 


6S« 


220 


SINES  AND  TANGENTS. 


23< 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

2.5 

26 

27 

28 

29 

m 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 


iPP 


9.57a575 
573888 
574200 
574512 
574824 
575136 
575447 
575758 
576069 
576379 
576689 

9.57G999 
577309 
577618 
577927 
678236 
57&515 
5788.53 
579162 
579470 
5797 

9..580085 
580.392 


581005 
581312 
581618 
581924 
582229 
582535 
582840 

9.583145 
583149 
583754 
5840.'i8 
581361 
581665 
681968 
585272 
585571 
685877 

9.5«6179 
586482 
586783 
587085 
587386 
587688 
587989 
588289 


588890 
}.  589190 
589189 
589789 
590088 
590387 
590686 
590984 
591282 
591.580, ,  ^„ 
591878  "^ 


I 

5.21 
5.20 
5.20 
5.20 
5.20 
5.19 
.5.18 
5.18 
5.17 
.5.17 
5.17 
5.16 
5.15 
5.15 
.5.15 
.5.15 
.5.14 
5.13 
.5.13 
.5.13 
5.12 
.5.12 
.5.12 
.5.11 
.5.11 
5.10 
.5.10 


5.08 
5.08 
5.08 
5.07 
5.07 
.5.06 
5.06 
5.05 
5.05 
5.05 
5.04 
5.04 
5.03 
5.03 
5.03 
5.02 
5.02 
5.01 
■5.01 
.5.01 
5.00 
5.00 
4.99 
4.99 
4.98 
4.98 
4.98 
4.97 
4.97 
4.97 


9.606410 
606773 
607137 
607500 
607863 
60822,5 
608588 
608950 
609312 
609674 
610036 

9.61039 
6107.59 
611120 
611180 
611841 
612201 
612-561 
612921 
613281 
613641 

J.  614000 
614^39 
614718 
615077 
615i:S5 
615793 
616151 
616509 
616867 
617224 

).617582 
617939 
618295 
6186.52 
610008 
619364 
619720 
620076 
620132 
620787 

1.621142 
621197 
621852 
622207 
622.561 
622915 
623269 
623623 
62.3976 
624,330 

1.621683 
62.50.36 
625;388 
62.5741 
626093 
626445 
626797 
627149 
627501 
627852 


PPl"  _3L 
60 
59 
58 
57 
56 
&5 
54 
53 


o.»0 

5.86 
5.85 


Cosine.   I  PPl"    Cotang.    PPl 


9.591878 
592176 
692473 
692770 
693067 
693363 
693659 
593955 
694251 
591547 
594812 

9.595137 
69.5132 
695727 
596021 
690315 
596609 
596903 
597196 
597490 
697783 

9.598075 
698;3(J8 
698660 
698952 
699244 
699536 
599827 
600118 
600109 
600700 
600990 
601280 
601570 
601860 
C021.50 
002439 
602728 
603017 
603305 
603594 

9.603882 
601170 
6044.57 
601745 
60.5032 
605.319 
605606 
605892 
606179 
606465 

9.606751 
6070^^6 
607322 
607607 
607892 
608177 


608745 


609313 


4.96 
1.95 
4.95 
4.95 
4.91 
4.94 
4.93 
4.93 
4.93 
4.92 
4.92 
4.92 
4.91 
4.90 
4.90 
4.90 
4.90 
4.89 
4.89 
4.88 
4.88 
4.87 
4.87 
4.87 
4.86 
4.86 
4.85 
4.85 
4.85 
4.8.5 
4.84 
4.83 
4.83 
4.83 
4.83 
4.82 
4.82 
4.82 
4.81 
4.81 
4.80 
4.80 
4.79 
4.79 
4.79 
4.78 
4.78 
4.78 
4.77 
4.77 
4.76 
4.76 
4.76 
4.75 
4.75 
4.75 
4.74 
4.74 
4.73 
4.73 


9.627852 


628554 
628905 
629255 
629606 
629956 
630306 
6;:?06,56 

6;^ioa5 

6313,55 
9.631704 
6320.53 
6,32402 
6327.50 


633447 
63379.5 
634143 
6.34490 
6348;» 
9.6a5185 
6;35.532 
635879 
636226 
636572 


637265 
637611 
6.37956 
638302 

9.6,38647 
638992 
639337 
639682 
640027 
640371 
640716 
&11060 
641404 
641747 

9.&42091 
642134 
642777 
643120 
-613463 
643806 
&14148 
614490 
644832 
645174 

9.645516 
645857 


646540 


647222 
647562 
647903 
648243 
6485831 


.5.85 
.5.85 
5.85 

.5.84 
5.84 

5.8:3 

.5.83 
5.83 
5.83 
5.82 
5.82 
5.82 
5.81 
,5.81 
.5.81 
5.80 
5.80 
5.80 
5.79 
,5.79 
5.79 
.5.78 
5.78 
5.78 
5.77 
5.77 
5.77 
5.77 
5.76 
5.76 
5.76 
5.75 
5.75 
5.75 
5.74 
5.74 
5.74 
5.73 
5.73 
5.73 
5.72 
5.72 
5.72 
5.72 
5.71 
5.71 
6.71 
5.70 
5.70 
5.70 
5.69 
5.69 
5.69 
5.69 
5.68 
5.68 
6.68 
5.67 
5.67 
5.67 


Cosine.   I  PPl" I  Cotaiis.  |PPl"    M 


60 
59 
58 
57 
56 
.55 
54 

5;^ 

52 
51 
50 
49 
48 
47- 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
.32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 


61  < 


383 


66' 


24" 


TABLE   IV.— LOGARITHMIC 


25 « 


650 


9.60931; 


610164 
610447 
610729 
611012 
611294 
611576 
6118,58 
612140 

9.612421 
612702 
612983 
613264 
6ia545 
6i;3825 
614105 
614385 
614665 
614914 

9.615223 
615502 
615781 
616060 
616338 
616616 
616891 
617172 
617450 
617727 

9.618004 
618281 
618.558 
618834 
619110 
619386 
619662 
619938 
620213 
620488 

9.620763 
621038 
621313 
621587 
621861 
622ia5 
622409 


622956 
623229 
.62a502 
62:3774 
624047 
624319 
624591 
624863 
62.5135 
625406 
625677 


Coainf.  PPl 


PI' 


4.73 
4.72 
4.72 
4.72 
4.71 
4.71 
4.70 
4.70 
4.70 
4.70 
4.69 
4.69 
4.68 
4.68 
4.67 
4.67 
4.67 
4.66 
4.66 
4.66 
4.65 
4.65 
4.65 
4.64 
4.64 
4.64 
4.63 
4.63 
4.62 
4.62 
1.62 
4.61 
4.61 
4.61 
4.60 
4.60 
4.60 
4. .59 
4.59 
4.59 
4..58 
4..58 
4..57 
4.57 
4.57 
4.-56 
4.56 
4..56 
4.55 
4.55 
4..55 
4..54 
4.54 
4.54 
4..53 
4.53 
4.53 
4.52 
4.52 
4.52 


Taiij?.   PPl"  M 


.6485^3 
648923 
649263 
649602 
649942 
650281 
6.50620 


651297 
6516.36 
651974 

9.6.52312 
6526.30 
6.52988 
653;}26 
653663 
654000 
654;J37 
654674 
6-55011 
655348 

9.6.55684 
656020 
656a56 
6.56092 
657028 
657364 
6.57699 
658034 
658:369 
658704 

9.659039 
659373 
659708 
660042 
660376 
660710 
661043 
661377 
661710 
662043 

9.662:376 
662709 
66:3042 
663375 
663707 
664039 
664:371 
664703 
6650a5 
665;3(J6 


5.58 


666029 
666:360 
660691 
6(57021 
667:352 
667682 
6()8013 
668:34:3 
668673 


CotRTIK.  PPP'  M 


9.625948 
626219 
626490 
626700 
627030 
627300 
627570 
627840 
628109 
628378 
628647 

9.628916 
629185 
629453 
629721 
629989 
630257 
630524 
6,S0792 
631059 
631:326 

9.631.593 
631859 
632125 
632392 
632658 
632923 
638189 
633454 
633719 
6a3984 

9.634249 
634514 
634778 
635042 
635306 
635570 
635834 
636097 
636360 
636623 

9.636886 
637148 
6:37411 
637673 
6:379a5 
638197 
638458 
638720 
638981 
6?39242 

9.639503 
639764 
640024 
640284 
640544 
640804 
^41064 
641324 
641583 
641»42 


Cosine.    PPl" 


PIM"!     Tiiiig. 

).  66^73 

669002 

669a32 

669661 


4.51 
4.51 
4.50 
4.50 
4.50 
4.50 
4.50 
4.49 
4.49 
4.48 
4.48 
4.47 
4.47 
4.47 
4.46 
4.46 
4.46 
4.46 
4.45 
4.45 
4.45 
4.44 
4.44 
4.44 
4.43 
4.43 
4.43 
4.42 
4.42 
4.42 
4.41 
4.41 
4.40 
4.40 
4.40 
4.40 
4.39 
4.39 
4.38 
4.38 
4.38 
4.37 
4.37 
4.37 
4.37 
4.36 
4.36 
4.36 
4.35 
4.35 
4.35 
4.34 
4.J34 
4.a4 
4.a3 
4.33 
4.33 
4.32 
4.32 
4.32 


670320 
670649 
670977 
671306 
671635 
671963 

9.672291 
672619 
672947 
678274 
673()02 
67,3929 
674257 
674,584 
674911 
675237 

9.67,5564 
67.5890 
676217 
676543 
676869 
677194 
677,520 
677846 
678171 
678496 

9.678821 
679146 
679471 
679795 
680120 
680444 
680768 
681092 
681416 
681740 

9.682063 
682387 
682710 
683033 
68335() 
683679 
684001 
684324 
684646 
684J)68 

9.685290 
685612 
6859;M 
68()25.5 
686577 
686898 
687219 
687540 
687861 
688182 


PP 


5.50 
5.49 
5.49 
5.49 
5.48 
5.48 
5.48 
5.48 
5.47 
5.47 
5.47 
5.47 
,5.46 
,5.46 
5.46 
5.46 
5.45 
5.45 
5.45 
5.44 
5.44 
5.44 
5.44 
.5.43 
5.43 
5.43 
5.43 
5.42 
5.42 
5.42 
5.42 
5.41 
5.41 
5.41 
5.41 
5.40 
5.40 
5.40 
5.40 
5.39 
5.39 
5.39 
5.39 
5.38 
5.38 
5.38 
5.38 
5.37 
5.37 
5.37 
5.37 
5.36 
5.:36 
5.36 
,5.36 
5.35 
5.35 
5.35 
5.35 
5.34 


Cotang.  PPl"  M 


384 


64' 


26" 


SINliS  AND  TANGENTS. 


270 


8n 


0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
34 
2-5 
26 
27 
28 
29 
30 
31 
32 
33 
34 
So 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


,641842 
642101 
642360 
642618 
642877 
643135 
643393 
64:36.50 
643908 
644ia5 
644423 
.644680 
644936 
645193 
64r>4;50 
645708 
645962 
646218 
64(5474 
616729 
646981 
.617240 
647494 
647749 
648001 
648258 
618512 
648766 
649020 
649274 
649527 
.619781 
050034 
6^30287 
650539 
650792 
651014 
651297 
651519 
651800 
652052 
.652301 
652555 
652806 
653057 
653308 
65a558 
653808 
&54059 
654309 
654558 
1.654808 
655058 
655307 
655556 
655805 
656054 


I' PI" 


Tans.      I'1>I"    M 


656551 
656799 
6.57047 


31 
4.31 

30 
4.30 
4.;30 
4.-30 
4.30 
4.29 
4.29 
4.29 
4.28 
4,28 
4.28 
4.27 
4.27 
4.27 
4.26 
4.26 
4.26 
4.2-5 
4.25 
4.'M 
4.24 
4.24 
4.23 
4.23 
4.23 
4.23 
4.23 
4.22 
4.22 
4.22 
4.22 
4.21 
4.21 
4.21 
4.20 
4.20 
4.20 
4.19 
4.19 
4.18 
4.18 
1.18 
4.18 
4.17 
4.17 
4.17 
4.17 
4.16 
4.16 
4.16 
4.16 
4.15 
4.15 
4.15 
4.14 
4.14 
4.14 
4.13 


Cosino, 


9.688182 
688.502 
688823 
689143 
689463 


690103 
690423 
690742 
691062 
691381 
.691700 
692019 
6923:38 
692&56 


693293 
69,3612 
6939:30 
694248 
694.566 
.694883 
69.5201 
69.5518 


6961.53 
696470 
696787 
697103 
697420 
697736 
.6980-5:3 


699001 
699316 
6996:32 
699947 
7002(33 
700-578 
700893 

9.701208 
701523 
701837 
702152 
702466 
702781 
70:3095 
703409 
70-3722 
704036 

9.704:3-50 
70466:5 
704976 
705290 
70^5603 
705916 
706228 
706541 
7068,54 
707166 


5.34 
5.:^ 
5.-34 
-5.33 
5.33 
-5.a3 
.5.33 
5.33 
5.32 
5.32 
5.32 
5.31 
5.-31 
5.31 
5.31 
5.31 

.30 
,5.30 
.5.30 
5., 30 
.5.29 

.29 
5.29 
,5.29 


Siiip. 


9.6,57047 
657295 
657,542 
657790 
6580,37 
658284 
658531 
6,58778 
659025 
659271 
ft59517 

9.ft597()3 
660009 
6602-55 
66a501 
660746 
660991 
6612:3() 
661481 
661726 
661970 

9.6()2214 
662459 
66270:3 


I'Pr 


663133 
663(577 
663920 
6641(53 
664406 

9. 664(548 
664891 
6651:3:3 
66,5;{75 
66.5617 
66;58,59 
066100 
666:342 
666583 
666824 

9. 6(57065 
067:305 
667,546 
667786 
668027 
6(582(57 
66&506 
668746 
668986 
6(5922-5 

9.669464 
669703 
669942 
670181 
670419 
670658 
670896 
671134 
671372 
671609 


4.13 
4.13 
4.12 
4.12 
4.12 
4.12 
4.11 
4.11 
4.11 
4.10 
4.10 
4.10 
4.09 
4.09 
4.09 
1.09 
4.08 
4.08 
4.08 
4.07 
4.07 
4.07 
4.07 
4.06 
4.06 
4.06 
4.a5 
4.05 
4.a5 
4.05 
4.04 
4.04 
4.04 
4.03 
4.03 
4.03 
4.02 
4.02 
4.02 
4.02 
4.01 
4.01 
4.01 
4.01 
4.00 
4.00 
4.00 
3.99 
3.99 
3.99 
3.99 
3.98 
3.98 
3.98 
3.97 
3.97 
3.97 
3.97 
3.96 
3.96 


PP1"|  Cotans.    PPl"    M.      M.      Cosino.     PPl"    Cotang.    PPl"    M 


ins. 


9.7071(56 
707478  -• 
707790  '^ 
708102 
708414 
708726 
7090:37 
709:349 
709660 
709971 
710282 

9.710-593 
710904 
711215 
711.52.5 
7118:36 
712146 
7121.56 
712766 
71:3076 
71:3:38(5 

9.71-369(5 
714005 
714:314 
714624 
714933 
715242 
715551 
71,5860 
716168 
716477 

9.71(578-5 
717093 
717401 
717709 
718017 
718:32-5 
71863:3 
718940 
719248 
719-555 

9.719862 
720169 
720476 
72078:3 
721089 
721:39(5 
721702 
722009 
722315 
722621 

9.722927 
723232 
72:3538 
72:3844 
724149 
724451 
724760 
72,5065 
725370 
725674 


63" 


Trig.— 33. 


385 


6*i« 


28  « 


TABLE   IV.— LOGARITHMIC 


29« 


0 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 


.Sine 


9.671609 
671847 
672084 
672321 
672558 
672795 
673032 
673268 
673505 
673741 
673977 

9.674213 
674448 
674684 
674919 
675155 
675390 
675624 
675859 
676094 
676328 

9.676562 
676796 
677030 
677264 
677498 
677731 
677964 
678197 
678430 
678663 


Pin' 


679128 
679360 
679592 
679824 
680056 
680288 
680519 
680750 


9.681213 
681443 
681674 
681905 
682135 
682365 
682,595 
682825 
683055 
683284 

9.683514 
683743 
683972 
684201 
684430 
684658 
684887 
68.5115 
68.5343 
685571 


(JositK 


3.96 
3.95 
3.95 
3.95 
3.95 
3.94 
3.94 
3.94 
3.94 
3.93 
3.93 
3.93 
3.92 
3.92 
3.92 
3.92 
3.91 
3.91 
3.91 
3.91 
3.90 
3.90 
3.90 
3.90 
3.89 
3.89 
3.89 
3.88 
3.88 
3.88 
3.88 
3.87 
3.87 
3.87 
3.87 
3.86 
3.86 
3.86 
3.85 

3.a5 

3.85 
3.85 
3.84 
.84 
3.84 
3.84 
3.83 
3.83 
3.83 
3.83 


3.82 
3.81 
3.81 
.81 
3.80 
3.80 
3.80 


TaiiK.   m 


9.725674 
725979 
726284 
726588 
726892 
727197 
727501 
727805 
728109 
728412 
728716 

9.729020 
729323 
729626 
729929 
730233 
7305a5 
730838 
731141 
731444 
731746 

9.732048 
732351 
7.32653 
732955 
73.3257 
733558 


734162 
7.34463 
734764 
9.735066 
735367 
735668 
735969 


736570 
736870 
737171 
737471 
737771 

9.738071 
738371 
738671 
738971 
739271 
739570 
739870 
740169 
740468 
740767 

9.741066 
741365 
741664 
741962 
742261 
742559 
742858 
743156 
743454 
743752 


PPl"  CotariK.  PPl"  M 


5.08 
5.07 
5.07 
5.07 
5.07 
5.07 
5.06 
5.06 
5.06 
5.06 
5.06 
5.05 
5.05 
5.05 
5.05 
5.05 
5.04 
5.04 
5.04 
5.04 
5.04 
5.03 
5.03 
5.03 
5.03 
5.03 
5.02 
5.02 
.5.02 
5.02 
5.02 
5.02 
5.01 
5.01 
5.01 
5.01 
5.01 
5.00 
5.00 
5.00 
5.00 
5.00 
4.99 
4.99 
4.99 
4.99 
4.99 
4.99 
4.98 
4.98 
4.98 
4.98 
4.98 
4.97 
4.97 
4.97 
4.97 
4.97 
4.97 


31.   M 

60 
59 
58 
57 
56 
55 
54 
53 
52 
61 
50 
49 
48 
47 
4Q 
45 
44 
43 
42 
41 
40 


9.685,571 
685799 
686027 
686254 
686482 
686709 
686936 
687163 


687616 
687843 


688521 


688972 
689198 
689423 
689648 


M. 


690098 
9.690323 
690548 
690772 
690996 
691220 
691444 
691668 
691892 
692115 
692339 
9.692562 
692785 
693008 
69,3231 
693453 
693676 
693898 
694120 
694;S42 
694564 
9.694786 
695007 
695229 
695450 
695671 


696113 
696334 
696554 
696775 
9.696995 
697215 
6974,35 
697654 
697874 


PPl' 


698313 
698532 
698751 


3.80 
3.79 
3.79 
3.79 
3.79 
3.78 
3.78 
3.78 
3.78 
3.77 
3.77 
3.77 
3.77 
3.76 
3.76 
3.76 
8.76 
3.75 
3.75 
3.75 
3.75 
3.74 
3.74 
3.74 
3.74 
3.73 
3.73 
3.73 
3.73 
3.72 
3.72 
3.72 
3.71 
3.71 
3.71 
3.71 
3.70 
3.70 
3.70 
3.70 
3.69 
3.69 
3.69 
3.69 
3.68 
3.6» 
3.68 
3.68 
3.67 
3.67 
3.67 
.67 
3.66 
3.66 
3.66 
3.66 
3.65 
3.65 
3.65 
3.65 


ang. 


9.743752 
7440,50 
744348 
744645 
744943 
745240 
74,55.38 
7458a5 
746132 
746429 
746726 

9.747023 
747319 
747616 
747913 
748209 
748505 
748801 
749097 
749393 
749689 

9.749985 
750281 
750576 
750872 
751167 
751462 
751757 
752052 
752347 
752642 

9.752937 
753231 
753526 
753820 
754115 
754409 
754703 
754997 
755291 
755585 

9.7.55878 
756172 
756465 
756759 
7.57052 
757345 
757638 
757931 
758224 
758517 

).  758810 
759102 
759395 
759687 
759979 
760272 
760564 
7()0856 
761148 
761439 


PPl 


4.96 
4.96 
4.96 
4.96 
4.96 
4.96 
4.95 
4.95 
4.95 
4.9(5 
4.95 
4.94 
4.94 
4.94 
4.94 
4.94 
4.93 
4.93 
4.93 
4.93 
4.93 
4.93 
4.92 
4.92 
4.92 
4.92 
4.92 
4.92 
4.91 
4.91 
4.91 
4.91 
4.91 
4.91 
4.90 
4.90 
4.90 
4.90 
4.90 
4.90 
4.89 
4.89 
4.89 
4.89 
4.89 
4.89 
4.88 
4.88 
4.88 
4.88 
4.88 
4.88 
4.87 
4.87 
4.87 
4.87 
4.87 
4.87 
4.86 


Cosine.     PPl"|  Cotantr.    PPl"    M 


ev 


386 


60< 


30' 


SINES  AND  TANGENTS. 


31< 


.698970 


699626 
699814 
700062 
700280 
700498 
700716 
700933 
701151 

9.701368 
70158.5 
701802 
702019 
7022:36 
702452 
702669 
70288.5 
703101 
703317 

9.70a5*i 
703749 
703964 
704179 
704395 
704610 
704825 
70.5040 
70.5254 
705469 

9.70.56aS 
705898 
706112 
706326 
7065:39 
7067.53 
7069(37 
707180 
707393 
707606 

9.707819 
7080:?2 
708245 
7084.58 
708670 
708882 
709094 
709.:]06 
709518 
7097:30 

9.709941 
71015:3 
710364 
710,575 
710786 
710997 
711208 
711419 
711629 
711839 


CosiiK 


PPI' 


3.65 
3.64 
3.64 
3.&1 
3.63 
3.63 
3.63 
3.63 
3.63 
3.62 
3.62 
3.62 
3.62 
3.62 
3.62 
3.61 
3.61 
3.60 
3.60 
.3.60 
3.60 
3.60 
3.59 
3.59 
3.59 
3.58 
3.58 
:3.58 
3.58 
3.58 
3.57 
3.57 
3.57 
3.57 
3.56 
3.56 
3.56 
3.55 
;3.55 
:3.55 
3.55 
3.55 
;3.55 
3.54 
3.54 
3.53 
;3.5;3 
.3..53 
3.53 
:3..53 
3..52 
3.52 
3.52 
3.52 
3.52 
3.52 
3.51 
3.. 51 
3.50 
3.50 


PPl" 


Taiu 


9.761439 
761731 
762023 
762314 


762897 
763188 
763479 
763770 
764061 
764:3.52 

9.764643 
7649:33 
76.5224 
765.514 
76.5805 
766095 
766:385 
766675 
76696.5 
76725.5 

9.767545 
767834 
768124 
768414 
768703 


769.571 


770148 
9.7704.37 
770726 
771015 
771303 
771.592 
771880 
772168 
772457 
772745 
773033 
9.773321 
773608 
773896 
774184 
774471 
774759 
775ai6 
775333 
775621 
775908 
9.77619.5 
776482 
776768 
777055 
777342 
777628 
777915 
778201 
778488 
778774 


Cotaiu 


PPI"  M 


4.86 
4.86 
4.86 
4.86 
4.85 
4.85 
4.85 
4.8.5 
4.85 
4.85 
4.85 
4.84 
4.84 
4.84 
4.84 
4.84 
4.83 
4.83 
4.83 
4.83 
4.8:3 
4.83 
4.83 
4.82 
4.82 
4.82 
4.82 
4.82 
4.82 
4.81 
4.81 
4.81 
4.81 
4.81 
4.81 
4.80 
4.80 
4.80 
4.80 
4.80 
4.80 
4.80 
4.79 
4.79 
4.79 
4.79 
4.79 
4.79 
4.79 
4.78 
4.78 
4.78 
4.78 
4.78 
4.78 
4.78 
4.77 
4.77 
4.77 
4.77 


PPI" I  M. 


M. 


60 


9.7 


.71 
712050 
712260 
IVMiQ 
712679 


713098 
713308 
713517 
71:3726 
71:39a5 
.714144 
714:3.-)2 
714.5IU 
714769 
714978 
715186 
71.5:394 
7ir)602 
715809 
71601 
.716224 
71f>432 
7166:39 
716)^6 
717a5:3 
7172.59 
71740() 
717( 
717879 
71808.5 
.718291 
718197 
718703 
718909 
719114 
719:320 
71ft52.5 
7197:30 
7199.35 
720140 
.720:345 
720.549 
720754 
7209.58 
721162 
721366 
721570 
721774 
721978 
722181 
,722385 
722588 
722791 
722994 
723197 
72:3400 
723603 
723805 
724007 
724210 


PPI' 


3.50 
3.50 
3.50 
3.50 
3.50 
3.49 
3.49 
3.49 
3.48 
3.48 
3.48 
3.48 
3.47 
:3.47 
3.47 
3.47 
3.47 
3.46 
:3.46 
3.46 
3.46 
3.45 
3.45 
3.45 
3.45 
3.45 
3.44 
3.44 
3.44 
3.43 
3.43 
3.43 
;3.43 
3.43 
3.43 
3.42 
:3.42 
13.42 
3.42 
3.41 
3.41 
3.41 
:3.41 
3.40 
;3.40 
.40 
3.40 
3.40 
3.40 

3.:39 
3.39 
3.39 
3.38 
3.38 
3.38 


3.37 

3.37, 

37 


Cosino.   |PPl"    Cotang 


laiiK. 


.778774 
779060 
779346 
779632 
779918 
780203 
780489 
780775 
781060 
78iai6 
781&31 

.781916 
782201 
782486 
782771 
783056 
783341 


PPI"  M. 


783910 
784195 
781479 

'.7847(>1 
785048 
785332 
785616 
785900 
786184 
786468 
786752 
787036 
787319 

.78760:3 
787886 
788170 
7884,53 


789302 
789585 
789868 
790151 

.790434 
790710 
790999 
791281 
79156:3 
791846 
792128 
792410 
792692 
792974 

.793256 
793538 
793819 
794101 
794383 
794664 
794946 
79.5227 
79.5508 
79.5789 


69 


59" 


387 


5S< 


S'Z' 


TABLE   IV.— LOGARITHMIC 


33" 


51° 


Tautr.    IPr 


9.724210 

724412 

724614 

724816 

725017 

725219 

725420 

725622 

725823 

726024 

726225 
9.726426 

726626 

726827 

727027 

727228 

727428 

727628 

727828 

728027 

728227 
9.728427 

728626 

72882.5 

729024 

729223 

729422 

729621 

729820 

730018 

730217 
9.730415 

730613 

7;^811 

731009 

731206 

731404 

731602 1 

7317991 

731996! 

732193' 
9.732390 

732587 

732781 

732980 

733177 

733373 

73a569 

73;^65 

73;S961 

7341571 
9.734353!.,  „. 

734.549  ^-^ 

7;34744  i'^. 

734939  ff. 

7.S5ia5  '  • - 

7a5525  l'^ 
7357191^-^ 

7:«9i4i:-^ 

736109  ^-"^^ 


3.;^ 

3.37 
3.37 
3.36 
3.36 
3.36 
3.35 
3.35 
3.a5 
3.35 
3.35 
3.34 
3.34 
3.34 
3.34 
3.33 
3.33 
3.33 
3.33 
3,33 
3.33 
3.32 
3.32 
3..S2 
3.32 
3.32 
3.31 
3.31 
3.31 
3.30 
3.30 
3.. 30 
3.30 
3.;30 
3.29 
3.29 
3.29 
3.29 
3.29 
3.28 
3.28 
3.28 
3.28 
3.28 
3.27 
3.27 
3.27 
3.27 
3.27 
3.26 
3.26 


9. 795789 j 
796070 
7963.51 
796632 
796913 
797194 
797474 
797755 
7980.36 
798316 


9.798877 
799157 
7994.37 
799717 
799997 
800277 
800,557 
800836 
801116 
801.396 

9.801675 
8019.55 
802234 
802.513 
802792 
803072 
803351 


803909 
804187 
9.804466 
804745 
80502:3 
805302 
80.5580 
805859 
806137 
806415 


80(5971 

).  807249 

807.527 

807805 


808361 
808638 


809193 
809471 
809748 

9.810025 
810302 
810580 
81085' 
811134 
811410 
811687 
811964 

•  812241 
812,517 


4.68 
4.()8 
4.68 
4.68 
4.68 
4.68 
4.68 
4.68 
4.67 
4.67 
4.67 
4.67 
4,67 
4.67 
4.07 
4.67 
4.66 
4.66 
4.66 
4.66 
4.66 
4.66 
4.66 
4.6.5 
4.65 
4.65 
4.65 
4.65 
4.65 
iM 
4.60 
4,65 
4.64 
4.64 
4.64 
4.64 
4.64 
4.64 
4,63 
4,63 
4.63 
4.63 
4.63 
4.63 
4.63 
4.63 

4.6:3 

4.62 
4.62 
4.62 
4.62 
4.62 
4.62 
4.62 
4.62 
4.60 
4.61 
4.61 
4.61 
4.61 


CoHine.   I  l'JM"|  Cotang.  !  PIM 


M. 


9.736109 
736:303 
736498 
73()692 
736886 
737080 
7:37274 
737467 
737661 
737855 
738048 

9.738241 
738434 
738627 
738820 
739013 
739206 


739590 
739783 
739975 

9.740167 
740359 
740550 
740742 
740934 
741125 
741,316 
741508 
741699 
741 

9.742080 
742271 
742462 
742652 
742842 
743033 
74.3223 
743413 


IM'i" 


3.24 
3.24 
5.24 
3.23 
3.23 
3.23 
3.23 
3.23 
3.22 
3.22 
3.22 
3.22 
3.22 
3.22 
3.21 
3.21 
3.21 
3.21 
3.20 
3.20 
3.20 
3.20 
3.20 
3.19 
3.19 
3.19 
:3.19 
3.19 
3.18 
3.18 
3.18 
3.18 
3.18 
3.17 
3.17 
3.17 
3.17 
3.17 


74.3602^-J^ 
74:3792!^-^ 
74.qt|8'2  ^'^^ 


744171 
744361 
744.350 
744739 
744928 
745117 
745306 
745494 
745683 
9.745871! 


:3.16 

3.16 

3.15 

3.15 

3.15 

3.15 

3.15 

3.14 

3.14 

3.14 

3  14 
7460601 J  ;* 

7462481^  ,. J 
746436!  ^-f^ 
7466241^;^ 
746812if}^ 
746999  !*^-;5 
/4/187  Q  ,„ 
747374 i^-J^ 
747562p^ 

osiiic.  PIM' 


Tni.K. 


9.812.517 
812794 
813070 
8i;]347 
813623 
813899 
814176 
814452 
814728 
815001 
815280 

9.815555 
81.58:31 
816107 
816382 
8166.58 
816933 
817209 
817484 
817759 
818035 

9.818310 
818585 
818860 
819135 
819410 
819684 
819959 
8202,34 
820508 
820783 

9.8210.57 
821332 
821606 
821880 
8221.54 
822429 
822703 
822977 
823251 
823524 

9.823798 
824072 
824:345 
824619 
824893 
825166 
825439 
825713 
825986 
826259 

9.826532 
826805 
827078 
827351 
827624 
827897 
828170 
828442 
828715 
828987 


Cotang.  PPl"  M 


388 


56' 


34° 


SINES  AND  TANGENTS. 


35° 


Siiif 


.747.^2 
747749 
747936 
74812  J 
748;U0 
748497 
7486a3 
748870 
749056 
749213 
749429 

.749815 
749.S01 
749987 
750172 
7.50.3.58 
750543 
7.50729 
7.50914 
751099 
751281 

.751469 
751654 
751839 
752023 
752208 
752392 
752576 
752760 
752944 
753128 

.7,53:312 
753495 
75.3679 
753862 
754046 
754229 
754412 
7.54.595 
754778 
754960 

.755143 
75.5326 
75.5.508 
75.5690 
75.5872 
756051 
756236 
756418 
7.56600 
756782 

.756963 
757144 
757326 
757507 
757688 
757869 
7580.50 
7,58230 
75^11 
758591 


Cosine.     PPl 


FPl' 


3.12 
3.12 
3.12 
3.12 
3.11 
3.11 
3.11 
3.11 
^.11 
3.10 
3.10 
3.10 
3.10 
3.09 
3.09 
3.09 
3.09 
3.09 
3.08 
3.08 
3.08 
3.08 
3.08 
3.08 
3.07 
3.07 
3.07 
3.07 
3.07 
3.07 
3.08 
3.08 
3.06 
3.08 
3.05 
.3.05 
3.05 
3.05 
3.05 
3.01 
3.01 
3.01 
3.01 
3.01 
3.01 
3.03 
3.03 
3.03 
3.03 
3.03 
3.02 
3.02 
3.02 
3.02 
3.02 
3.02 
3.01 
3.01 
3.01 
8.01 


).  828987 
829260 
829.5:32 
8298a5 
8:30077 
8:30.349 
8.30621 
8:30893 
a3116.:> 
8314:37 
831709 

J.  8:31981 
832253 
8:32525 
832796 
8:33038 
8:3:3.3.39 
8:3:3611 
83-3882 
&311.54 
a34425 


8:34967 
8:352:38 
8135.509 
83.5780 
8.36051 
836.322 
8;36593 
836861 
837131 

1.837405 
8:37675 
8.37946 
8:38216 
8:38487 
838757 
839027 
839297 
8:39568 
8:39838 

1.810108 
840378 
840818 
810917 
811187 
8114,57 
841727 
841996 
842266 
842ai5 

).842.S05 
&4:3074 
843343 
84,3612 
843882 
844151 
844420 
844689 
844958 
845227 


FPl"  M 


4, 
1, 
4 

4 
4 
4 

4 

4 

4 

4 

4 

4 

4 

4 

4.48 

4.48 


Cotang.  PPl"  M 


,7.58.591 
7,58772 
7,58952 
759132 
759312 
759492 
759672 
759852 
760031 
760211 
760:390 

.76ft569 
760748 
7C0927 
761 10<) 
76128.5 
761464 
761W2 
761821 
761999 
762177 

.762:3.56 

7(i2rm 

762712 
762889 
76:3067 
763245 
76,3422 
76:3600 
763777 
76:3954 

.7&1131 
7(>i;308 
764485 
764662 
7(H8;38 
765015 
765191 
765367 
765544 
765720 

.765896 
766072 
766247 
766423 
766598 
766774 
760919 
767124 
767:300 
767475 

.767649 
7<)7824 
767999 
768173 
768348 
768522 
768697 
768871 
769045 
769219 


I'l' 


3.01 
:3.00 
:3.00 
:3.00 
3.00 
,3.00 
3.00 
2.99 
2.99 
2.99 
2.99 
2.98 
2.98 
2.98 
2.98 
2.98 
2.98 
2.97 
2.97 
2.97 
2.97 
2.97 
2.97 
2.96 
2.96 
2.96 
2.96 
2.96 
2.9.5 
2.95 
2.95 
2.95 
2.95 
2.9.5 
2.94 
2.M 
2.94 
2.94 
2.94 
2.93 
2.93 
2.93 
2.93 
2.93 
2.93 
2.92 
2.92 
2.92 
2.92 
2.92 
2.91 
2.91 
2.91 
2.91 
2.91 
2.90 
2.90 
2.90 
2.90 
2.90 


Tan 


9.84.5227 
845496 
845764 
8460:33 
846:302 
846570 
846839 
847108 
847376 
847644 
847913 

9.848181 
848449 
818717 
848986 
849254 
849522 
849790 
850057 
850325 
85059:3 

9.850861 
851129 
851.396 
851664 
851931 
852199 
852466 
8527:38 
85:3001 
85:32(58 

9.853585 
85:3802 
854069 
8iy3:3(i 
854<)03 
8^870 
855187 
85^04 
855()71 
8559:58 

9.856204 
856471 
856737 
857004 
857270 
857.537 
85780:3 
858069 
8583:36 
858602 

9.858868 
859iai 
859400 


859932 
860198 
8604&4 
860730 


861261 


55  « 


389 


Cosine.     PPl"    Cotang.    PPl"    M. 
__ 


36° 


TABLE   IV.— LOGARITHMIC 


3^0 


0 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
41 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 


M. 


r69219 


769740 


770087 
770260 
770433 
770606 
770779 
770952 

9.771125 
771298 
771470 
771643 
771815 
771987 
772159 
772331 
772503 
772675 

9.772847 
773018 
773190 
773361 
773533 
773704 
77;«75 
774046 
774217 
774388 

9.774558 
774729 
774899 
775070 
775240 
775410 
775580 
775750 
775920 
776090 

9.776259 
776429 
776598 
776768 
77693- 
777106 
777275 
777444 
777613 
777781 

9.777950 
778119 
77828- 
778455 
778624 
778792 
T78960 
779128 
779295 
779463 


PPl' 


C(>sin< 


2.90 

2.89 
2.89 
2.89 
2.89 
2.89 
2.88 
2.88 
2.88 
2.88 
2.88 
2.88 
2.87 
2.87 
2.87 
2.87 
2.87 
2.87 
2.87 
2.86 
2.86 
2.86 
2.86 
2.86 
2.85 
2.85 
2.85 
2.a5 
2,85 
2.85 
2.84 
2.84 
2.84 
2.84 
2.84 
2.84 

2.m 

2.83 
2.83 
2.83 
2.83 
2.83 
2.82 
2.82 
2.82 
2.82 
2.82 
2.82 
2.81 
2.81 
2.81 
2.81 
2.81 
2.80 
2.80 
2.80 
2.80 
2.80 
2.80 


•J'a 


J.  861261 
861527 
861792 
862a58 
862323 
862589 
8628W 
863119 
86338.5 
863650 
863915 

).  864180 
864445 
864710 
864975 
865240 
865505 
865770 
8660:35 


866564 


867094 
867358 
867623 
867887 
868152 
868416 


868945 
869209 

9.869473 
869737 
870001 
870265 
870529 
870793 
871057 
871321 
871585 
871849 

).  872112 
872376 
872640 
872903 
873167 
873430 
873694 
873957 
874220 
874484 

).  874747 
875010 
875273 
875537 
875800 
876063 
876326 
876589 
876852 
877114 


■IM" 


4.43 
4.43 
4.43 
4.42 
4.42 
4.42 
4.42 
4.42 
4.42 
4.42 
4.42 
4.42 
4.42 
4.42 
4.42 
4.41 
4.41 
4.41 
4.41 
4.41 
4.41 
4.41 
4.41 
4.41 
4.41 
4.41 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.39 
4.39 
4.39 
4.39 
4.39 
4.39 
4.39 

39 
4.39 

39 
4.39 

39 
4.39 
4.38 
4.38 

38 
4.38 


4.38 


PPl"l  Cotang.  IPIM"    M 


bine. 


9.779463 
7796;31 
779798 


780133 
780300 
780467 
780634 
780801 
780968 
781134 

9.7»1301 
781468 
781634 
781800 
781966 
782132 
782298 
782464 
782680 
782796 

9.782961 
783127 
783292 
783458 
783623 
783788 


784118 
784282 
784447 

).  784612 
784776 
784941 
785105 
785269 
7854:33 
785597 
785761 
785925 
786089 

1.786252 
786416 
786579 
786742 
786906 
787069 
787232 
787395 
787557 
787720 

'.787883 
788045 
788208 
788370 
788532 
788694 
788856 
789018 
789180 
789342 


PPl 


2.79 
2.79 
2.79 
2.79 
2.79 
2.78 
2.78 
2.78 
2.78 
2.78 
2.78 
2.78 
2.77 
2.77 
2.77 
2.77 
2.77 
2.77 
2.76 
2.76 
2.76 
2.76 
2.76 
2.75 
2.75 
2.75 
2.75 
2.75 
2.75 
2.75 
2.74 
2.74 
2.74 
2.74 
2.74 
2.73 
2.73 
2.73 
2.73 
2.73 
2.73 
2.73 
2.72 
2.72 
2.72 
2.72 
2.72 
2.72 
2.71 
2.71 
2.71 
2.71 
2.71 
2.71 
2.70 
2.70 
2.70 
2.70 
2.70 
2.70 


Tai 


9.877114 

877377 
877640 
877903 
878165 
878428 
878691 
878953 
879216 
879478 
879741 

9.880003 
880265 
880528 
880790 
881052 
881314 
881577 
8818:39 
882101 
882363 

9.882625 
882887 
883148 
883410 
883672 
883934 
884196 
88445' 
884719 
884980 

9. 88.5241 
885504 
885765 
886026 
886288 
886549 
886811 
887072 
887333 
887594 

9.887855 
888116 
888378 


4.38 
4.38 
4.38 
4.38 
4.38 
4.38 
4.38 
4.38 
4.37 
4.37 
4.37 
4.:37 
4.37 
4.37 
4.37 
4.37 
4.37 
4.37 
4.37 
4.37 
4.37 
4.36 
4.36 
4.36 
4.36 
4.36 
4.36 
4.36 
4.36 
4.36 
4.36 
4.36 
4.36 
4.36 
4.36 
4.36 
4.35 
4.35 
4.35 
4.35 
4.35 
4.35 
4.35 
4.35 
4.35 
4.35 
4.35 
35 
4.35 

890204  ^'f. 
9.890465|j  J 
890725  ^-^ 


889161 
889421 
889682 
889943 


PPl" 


891247 
891507 
891768 
892028 
892289 
892.549 
892810 


4.34 
4.34 
4.34 
4.34 
4.34 
4.34 
4.34 


53' 


390 


PPl"    Cotaiig.    PPl^^    M. 


38" 

M.     ! 


SINES  AND  TANGENTS. 


39" 


ppi' 


.789342 
789.304 

789827 


790149 
790310 
790471 
790632 
790793 
7909.>4 

.791115 
791275 
791436 
791596 
791757 
791917 
792077 
792237 
792:397 
792-W7 

.792716 
792876 
793035 
793195 
7933.54 
79a514 
79.3673 
7938.32 
793991 
794150 

.794.308 
794467 
794626 
794784 
794942 
795101 
7952.59 
79;>417 
795575 
7957*3 

.795891 
796049 
796206 
796;364 
79i;521 
796679 
7968:36 
796993 
797150 
797307 

.797464 
797621 
797777 
797934 


798247 
798403 
798560 
798716 

798872 


2.69 
2.69 
2.69 
2.69 
2.69 
2.69 
2.68 
2.68 
2.68 
2.68 
2.68 
2.68 
2.68 
2.67 
2.67 
2.67 
2.67 
2.67 
2.67 
2.66 
2.66 
2.66 
2.66 
2.66 
2.65 
2.65 
2.65 
2.65 
2.65 
2.65 
2.64 
2.64 
2.64 
2.64 
2.64 
2.64 
2.64 
2.63 
2.63 
2.63 
2.63 
2.63 
2.63 
2.63 
2.62 
2.62 
2.62 
2.£2 
2.62 
2.62 
2.62 
2.61 
2.61 
2.61 
2.61 
2.61 
2.61 
2.60 
2.60 
2.60 


PiM' 


Tanjr 


).  892810 
89:3070 
89:33:31 


89:38.51 
894111 
894:372 
894632 
894892 
8951.52 
89.5412 
.895672 
89.5932 
896192 
896452 
896712 
896971 
897231 
897491 
897751 


9.898270 


899827 
900087 
900346 
900605 

9.900864 
901124 
901383 
901642 
901901 
902160 
902420 
902679 
902938 
903197 

9.903456 
90.3714 
90:3973 
9042.32 
904491 
904750 
90.5008 
905267 
905526 
90.5785 

9.90<J043 
906:302 
906,560 
906819 
907077 
907336 
907594 
907853 
908111 


I'Pl" 


Cutaiig.  PPI"  M. 


M. 


M. 


SiM<>.    PPl" 


J.798872 
799028 
799184 
799339 
79949.5 
799651 
799806 
799962 
800117 
800272 
800427 

J.  800582 
800737 


801047 
801201 
801356 
801511 
801665 
801819 
801973 
.802128 
802282 
802436 


802743 
802897 
8a3a50 
803204 
80;3357 
803511 

9.80:3664 
803817 
803970 
804123 
804276 
804428 
804581 
804734 
801886 
8050:39 

9.805191 
805:^3 
8054S)t5 
805647 
805799 
803951 
806103 
806254 
806406 
806557 

9.806709 
806860 
807011 
807163 
807314 
807465 
807615 
807766 
807917 
808067 


Cosine. 


2.60 
2.60 
2.60 
2.60 
2.59 
2.59 
2.59 
2.59 
2.59 
2.58 
2.58 
2.58 
2.58 
2.58 
2.58 
2.58 
2.58 
2.57 
2.57 
2.57 
2..57 
2.57 
2.57 
2.56 
2.56 
2.56 
2.56 
2.56 
2.56 
2.55 
2.55 
2.55 
2.55 
2.55 
2.55 
2.54 
2.54 
2.54 
2M 
2.&4 
2.54 
2.54 
2,53 
2.53 
2.53 
2.53 
2.-53 
2.53 
2.53 
2.52 
2.52 
2.52 
2.52 
2.52 
2.52 
2.52 
2.51 
2.51 
2.51 
2.51 


PPl' 


Tai 


9.908.369 
90S628 
908886 
909144 
909402 
909660 
909918 
910177 
910435 


910951 

9.911209 
911467 
911725 
911982 
912240 
912498 
912756 
913014 
913271 
91:3529 

9.913787 
914044 
914302 
914.560 
914817 
915075 
915332 
91.5590 
915t«7 
916101 

9.916362 
916619 
916877 
917134 
917891 
917648 
917906 
918163 
91H420 
918677 

9.9189;i4 
919191 
919448 
919705 
919962 
920219 
920476 
920733 
920990 
921247 

9.921503 
921760 
922017 
922274 
922530 
922787 
923044 
923300 
923557 
923814 


Cotang. 


PPl' 


61^ 


391 


50* 


40^ 


TABLE    IV.— LOGAUITHMIC 


41 


.808067 
808218 
808368 
808519 


808819 


809119 
809269 
809419 
809569 

1.809718 
809868 
810017 
810167 
810316 
810465 
810614 
810763 
810912 
811061 

1.811210 
811358 
811507 
811655 
811804 
811952 
812100 
812248 
812396 
812544 

1.812692 
812840 
812988 
813ia5 
813283 
813430 
81*578 
81372.5 
813872 
814019 

>.814166 
814313 
814460 
814607 
814753 
814900 
8ir)(>46 
815193 
815:339 
81548.5 

).  815632 
815778 
815924 


816215 
816:361 
816507 
8166.52 
816798 
816943 


M.   Cosine. 


I'l' 


2.51 
2.51 
2.51 
2.50 
2.. 50 
2..50 
2..50 
2.  .50 
2.-50 
2.50 
2.49 
2.49 
2.49 
2.49 
2.49 
2.48 
2.48 
2.48 
2.48 
2.48 
2.48 
2.48 
2.48 
2.47 
2.47 
2.47 
2.47 
2.47 
47 
2.46 
2.46 
2.46 
2.46 
2.46 
2.46 
2.46 
2.46 
2.45 
2.45 
2.45 
2.4.5 
2.45 
2.45 
2.45 
2.44 
2.44 
2.44 
2.44 
2.44 
2.44 
2.43 
2.43 
2.43 
2.43 
2.43 
2.43 
2.43 
2.42 
2.42 
2.42 


»P1' 


rang.  PPl"  JM 


).  92:3814 
924070 
924327 
92458:3 
924840 
925096 
9253,52 
925609 
925865 
926122 
926378 

).  926634 
926890 
927147 
927403 
927659 
927915 
928171 
928427 
928684 
928940 

).  929196 
929452 
929708 
929964 
930220 
930475 
930731 
930987 
931243 
931499 

).9317;55 
932010 
9:32266 
932522 
932778 
933033 
933289 
9:33545 
933800 
934056 

).  934311 
934r}67 
934822 
935078 
935333 
935589 
9:35844 
936100 
936*55 
9:36611 

9.9:3686<» 
93712  L 
9373".  7 
937632 
937887 
938142 
938:398 
9386.53 


939163 


Cotang.  I  PPl"  M 


M. 


}>V 


9.816943 
817088 
817233 
817379 
817524 
817668 
817813 
817958 
818103 
818247 
818392 

9.818536 
818681 
81882') 
818969 
819113 
819257 
819401 
819545 
819689 
819832 

9.819976 
820120 
820263 
820406 
8205,50 
820693 
820836 
820979 
821122 
821265 

9.821407 
821550 
821693 
821835 
821977 
822120 
822262 
822404 
822546 
822688 

9.822830 
822972 
823114 
82:3255 
82:3397 
823539 
823680 
823821 


2.42 
2.42 
2.42 
2.42 
2.41 
2.41 
2.41 
2.41 
2.41 
2.41 
2.41 
2.41 
2.40 
2.40 
2.40 
2.40 
2.40 
2.40 
2.40 
2.39 
2.39 
2.39 
2.39 
2.39 
2.39 
2.38 
2.38 
2.38 
2.38 
2.38 
2.38 
2.38 
2.38 
2.37 
2.37 
2.37 
2.37 
2.37 
2.37 
2.37 
2.37 
2.36 
2.:36 
2.36 
2.36 
2.36 
2.36 
2.35 

12.35 
2.a5 
2.35 


8241041 
9.824245 
824,386 
824,527 
824668 
824808 
824949 
825090 
82,5230 
825371 
82-5511; 


2.;35 
2.34 
2.34 
2.34 
2.34 
2.34 
2.34 


Cosine.  PPl' 


Tiuig. 


9.9:39163 
939418 
9:39673 
939928 
94018:3 
940439 
940694 
940949 
941204 
94145fe 
941713 

9.941968 
94222:3 
9424' 
94273:3 
942988 
943243 
943498 
94:3752 
944007 
944262 

9.944517 
944771 
945026 
945281 
945535 
945790 
946045 
946299 
946554 
946808 

9.947063 
947318 
947572 
947827 
948081 
948335 
948590 
948844 
949099 
949353 

9.949608 
949862 
950116 
950.371 
950025 
950879 

9511:3:3 

"951388 
951642 
951896 
9.952150 
952405 
952659 
952913 
953167 
953421 
95.3675 
953929 
954183 
ft544;37 

Cotang. 


J.  PI, 


4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.2.5 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.23 
4.23 
4.23 
4.23 
4.28 


•PI" 


49' 


392 


48* 


42" 


SINES  AND  TANGENTS. 


43" 


Sine. 


31 

m 

37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 


9. 


825.511 
8256.51 
82-5791 
825931 
82tj071 
826211 
8263.51 
826491 
826631 
826770 
82<}910 
827049 
827189 
827328 
827467 
827606 
827745 
827884 
828023 
828162 
828301 
828439 
828.578 
828716 
82.S&5.5 
828993 
829131 
829269 
829407 
829.545 
82968;i 
829821 
8299.59 

s;mm 

830234 
8m372 
830.509 
8;}0646 
830781 
830921 
8.31058 
831195 
8513.32 
83U69 
8.3160J 
831742 
831879 
8.32015 
832152 
8;^2288 
832425 
.832;561 
832697 
8.328.33 
8.32969 
833105 
83,3241 
833377 
83,5512 
833648 


PPl"i     Tang. 


2.34 

2.m 
2.m 

2.3:3 
2.33 
2.a3 
2.33 
2.a3 
2.33 
2.-3.3 
2.32 
2.32 
2.32 
2.32 
2.-32 
2.32 
.-32 
2.32 
2.31 
2.31 
2.31 
2.. 31 
2.. 31 
2.31 
2.30 
2.30 
2.30 
2.30 
2.;30 
2.30 
2.30 
2.30 
2.29 
2.29 
2.29 
2.29 
2.29 
2.29 
2.29 
2.2S 
2,2,S 
2.2.S 
2.28 
2.2S 
2.28 
2.28 
2.28 
2.28 
2.27 
2.27 
2.27 
2.27 
2.27 
2.27 
2.27 
2.26 
2.26 
2.26 
2.26 
2.26 


9.9,54437 
9-54691 
9.54946 
955200 
9.554-54 
9.55708 
95.5961 
956215 
9-56469 
9.5672;3 
956977 

9.9-57231 
95748-5 
957739 
9-57993 
958247 
958500 
9587-54 


9-59516 

9.9-59769 
960023 
960277 
960.530 
960784 
9610:?8 
961292 
961545 
981799 
9620.52 

9.962306 
962.560 
962813 
96:5067 
96.3520 
963574 
96:5828 
961081 
96433.5 
964.588 

9.904842 
96-5095 
96;5;349 
96i5602 
9658.55 
9(56109 
966:362 
966616 
966869 
96712:3 

9.967:376 
967629 
967883 
968136 
968389 
968643 


969149 
969403 
969656 


PF 


M. 


Sine. 


PPl" 


9.83;3783 
833919 
8.340.54 
8:34189 
&3432.5 
834460 
834595 
834730 
8348&5 
834999 
835134 

9.835269 
83540:3 
835538 
835672 
835807 
835941 
836075 
836209 
8:36:343 
8:36477 

9.836611 
836745 
836878 
837012 
837146 
837279 
837412 
837->46 
8:37679 
8:37812 

9.837945 
838078 
838211 
8:38344 
8:38477 
838610 
838742 
838875 
839007 
839140 

9.839272 
8:59404 
8:39-536 
839668 
839800 
8:39932 
840064 
{340196 
840:528 
840459 

9.840591 
840722 
840854 
840985 
841116 
841247 
841378 
841509 
841640 
841771 


2.26 
2.25 
2.25 
2.25 
2.25 
2.25 
2.25 
2.25 
2.25 
2.25 
12.24 
'2.24 
2.24 
2.24 
2.24 
2.24 
2.24 
2.23 
2.2:3 
2.23 
2.23 
2.23 
2.23 
2.23 
:2.22 
2.22 
2.22 
2.22 
2-22 
2.22 
2.22 
2.22 
2.22 
2.21 
2.21 
2.21 
2.21 
2.21 
2.21 
2.21 
2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.19 
2.19 
2.19 
2.19 
2.19 
2.19 
2.19 
2.18 
2.18 
2.18 
2.18 
2.18 


4'y( 


Cosine.     PPl"    Cotang.    PPl"    M.      M.      Cosine.     PP 

393 


Tang. 


9.9()96.56 


970162 
970416 


970922 
971175 
971429 
971682 
9719.35 
972188 

9.972441 
972695 
972948 
973201 
9734.54 
97:3707 
97.3960 
974213 
974466 
974720 

9.974973 
975226 
975479 
9757:32 
975985 
976238 
976491 
976744 
976997 
977250 

9.977.503 
977756 


978262 
978515 
978768 
979021 
979274 
979527 
979780 

9.980033 
980286 
980,538 
980791 
981044 
981297 
981550 
981803 
982056 
982309 i 

9.982562 
982814 
983067 
983320 
983573 
983826 
984079 
984332 
984584 
984837 


Cotang.  PPl"  M 


31 


4«'^ 


394 


46' 


SINES  AND  TANGENTS. 


^-yc 


ISilK: 


PFl" 


9.&56934 
857056 
8,57178 
&57300 
857422 
8,57543 
8.5766.5 
857786 
857908 
858029 
8.58151 
9.858272 
858393 
8,58514 
858635 
858756 
8.58877 
8,58998 
859119 
8,592:39 
8.59360 
9.859480 
859601 
859721 
8.59842 
859962 
8()(X)82 
860202 
8()0322 
860442 
860562 
9.8<i0682 
860802 
860922 
861041 
861161 
861280 
861400 
861519 
S616;38 
0017.58 
9.861877 
861996 
862115 
8622;J4 
862;»i 
862471 
862.590 
862709 
862827 
862946 
9.863064 
86318;} 
863^501 
863419 
863538 
86ii656 
863774 
863892 
864010 
864127 


2.03 
03 
2.03 
2.0;^ 
2.03 
2.03 
2.02 
2.02 
2.02 
2.02 
2.02 
2.02 
2.02 
2.02 
2.02 
2.02 
2.01 
2.01 
2.01 
2.01 
2.01 
2.01 
2.01 
2.01 
2.00 
2.00 
2.00 
2.00 
2.00 
2.00 
2.00 
2.00 
1.99 
1.99 
1.99 
1.99 
1.99 
1.99 
1.99 
1.99 
1.98 
1.98 
1.98 
1.98 
1.98 
1.98 
1.98 
1.98 
1.98 
1.98 
1.97 
1.97 
1.97 
1.97 
1.97 
1.97 
1.97 
1.97 
1.97 
1.96 


Tans 


10.015163 
01.S116 
01.5668 
01.5921 
016174 
016427 
016680 
01693;3 
017186 
017438 
017691 

10.017944 
018197 
0184.50 
018703 


019209 
019462 
019714 
019967 
020220 

10.020473 
020726 
020979 
0212:32 
02148;5 
0217.38 
021991 
022244 
022497 
0227;50 

10.023003 
0232-56 
02:3509 
02:3762 
024015 
024268 
024.521 
024774 
025027 
02.5280 

10.02.>534 
02.5787 
026040 
026293 
026546 
026799 
027052 
027305 
027559 
027812 

10.028065 
028318 
02.^571 
028825 
029078 
029:331 
0295m 
029838 
030091 
030344 


PPI"1  M. 


4.21 
4.21 
4.21 
4.21 
4.21 
4.21 

21 
4.21 
4.21 
4.21 

21 
4.21 
4.21 

21 
4.21 
4.22 
4.22 
4.22 

22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 


(Cosine.  PPl"  Cotane.  PPl"  M 


|PPr'i  Tang.  I  PFl" 


9.8(>4127j 
864245 
864:363  i 
864481 
864.598 
864716 
8&483:3 
864950 
865068 
865185 
865302 

9.86,'5419 
865536 
865(>53 
865770 
86.5887 
866004 
866120 
866237 
866:35:3 
8(56470 

9.866586 
866703 
866819 


867a51 
867167 
867283 
867399 
867515 
8676:31 

9.867747 
867862 
867978 
868093 
868209 
868324 
868440 
868555 
868670 
868785 

9.868960 
869015 
869130 
869245 
869:360 
869474 
869.589 
869704 
869818 
86993:3 

9.870047 
870161 
870276 
870:390 
870504 
870618 
870732 
870846 
870960 
871073 


1.96 
1.96 
1.96 
1.96 
1.96 
1.96 
1.96 
1.95 
1.95 
1.95 
1.95 
1.95 
1.95 
95 
l.ft5 
1.95 
1.94 
1.94 
1.94 
1.94 

i.m 

1.94 
1.94 
1.94 
1.93 
1.93 
1.93 
1.93 
1.93 
1.93 
1.93 
1.9:3 
1.93 
1.92 
1.92 
1.92 
1.92 
1.92 
1.92 
1.92 
1.92 
1.92 
1.92 
1.91 
1.91 
1.91 
1.91 
1.91 
1.91 
1.91 
1.91 
1.90 
1.90 
1.90 
1.90 
1.90 
1.90 
1.90 
1.90 
1.90 


10.030:344 
0:50597 
0:50851 
031104 
031:3,57 
031611 
031864 
032117 
032,371 
032624 
032877 

10.0:33131 
03:3384 
0:336:38 
033891 
034145 
0.'y:398 
0:34651 
034905 
0a5158 
03,>412 

10.035665 
035919 
036172 
0.36426 
036680 
0:3693:3 
037187 
037440 
037694 
037948 

10.038201 
038455 
038708 
038962 
039216 
039470 
039723 
0:39977 
040231 
040484 

10.0107:38 
040992 
041246 
041500 
041753 
042007 
042261 
042515 
042769 
043023 

10.043277 
04*531 
04;37a5 
044039 
044292 
044546 
044800 
045054 
04.5309 
045563 


4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
.23 
4.23 
4.2:3 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.2:3 
4.23 
4.23 
4.23 
4.23 
4.23 
4.2:3 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.2:3 
4.2:3 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 


Cosine.     PPl"    Cotana.    PPl"    M. 


4a' 


395 


42« 


4S» 


TABLE  IV.— LO(iARlTHMlC 


49' 


M. 

4fF 


9.87107;} 
871187 
871301 
871414 
871528 
871641 
871755 
871868 
871981 
872095 
872208 

9.872-321 
872434 
872547 
872659 
872772 
872885 
872998 
873110 
873223 
■  873335 

9.873448 
873560 
873672 
873784 
873896 
874009 
874121 
874232 
874344 
874456 

9.874568 
874680 
874791 
874903 
87.5014 
875126 
8752;^7 
875348 
875459 
875571 

9.875682 
875793 
875904 
876014 
876125 
8762;56 
876347 
876457 
876.568 
876678 

9.876789 
876899 
877010 
877120 
877230 
877340 
877450 
877560 
877670 
877780 


I'Pl' 


Tans.    |1M»1"!  M. 


1.90 
1.89 
1.89 
1.89 
1.89 
1.89 
1.89 
1.89 
1.89 
1.88 
1.88 
1.88 
1.88 
1.88 
1.88 
1.88 
1.88 
1.88 
1.88 
1.87 
1.87 
1.87 
1.87 
1.87 
1.87 
1.87 
1.87 
1.87 
1.87 
1.86 
1.86 
1.86 
1.86 
1.86 
il.86 
1.86 
1.85 
1.8.5 
11.85 
1.85 
1.&5 
1.85 
1.85 
1.85 
l.S;3 
l.») 
1.&5 
1.84 
1.84 
1.84 
1.84 
1.84 
1.84 
1.84 
1.84 
1.83 
1.83 
1.8;^ 
1.83 
1.83 


Cosine. 


10.04.>563 
045817 
04(«)71 
04(532,5 
046579 
04683;^ 
047087 
047341 
047595 
047850 
048104 

10.048358 
048612 
0488(57 
049121 
049,375 
049629 
0198^1 
050i;38 
0,50392 
050647 

10.0-50901 
051156 
051410 
051665 
051919 
052173 
a52428 
052682 
052937 
0-53192 

10.053446 
053701 
0-539-55 
054210 
0544a5 
0.54719 
054974 
0-5.5229 
05,518;3 
0-557,38 

10.0-55993 
056248 
056.502 
056757 
057012 
0.5726' 
057522 
0.57 
0-58032 
058287 

10.058541 
058796 
059051 
059;}06 
059,561 
059817 
060072 
060,327 
060582 
060837 


4.23 
4.23 
4.23 
4.23 

4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.25 
4.2-5 
4.25 
4.25 
4:2,5 
4.2-5 
4.25 
4.23 
4.25 
4.a5 
4.25 
4.25 
4.2,5 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 


PIM"    Cotang.    PPl"    M. 


I'lM' 

i.as 

1.83 
1..S3 
1.83 
1.82 
1.82 
1.82 
1.82 
1.82 
1.82 
1.82 
1.82 
1.82 
1.82 
1.81 
1.81 
1.81 
1.81 
1.81 
1.81 
1.81 
1.81 
1.81 
1.80 
1.80 
1.80 
1.80 
1.80 
1.80 
1.80 
1.80 
1.80 
1.80 
1.79 
1.79 
1.79 
1.79 
1.79 
1.79 
1.79 
1.79 
1.79 
1.78 
1.78 
1.78 
1.78 

8829/ '  1  --0 
883084;, '^'2 
883191 1  ;*,1° 
9.88,32971  J'^ 
883404!!'^ 
883;j10 


9.877780 
877890 
877999 
878109 
878219 
878:528 
8784;:58 
87&547 
878656 
8787(56 
878875 

9.878981 
879093 
879202 
879311 
879420 
879529 
879637 
879746 
879855 
879963 

9.880072 
880180 
880289 
880397 
880505 
880613 
880722 
880830 
880938 
881046 

9.881153 
881201 
881369 
881477 
881584 
881692 
881799 
881907 
882014 
882121 

9.882229 
882^«6 
882443 
8825i')0 
882&17 
882764 


88:5617 
883723 
88.3829 
883936 
884042 
881148 
884254 


M.     Cosine.    PPl 


1.77 
1.77 
1.77 
1.77 
1.77 
1.77 
1.77 


Tiing. 
10.()()08;37 
061092 

0(;i;547 


061(i02|'' 
061K58!^' 
062113 


062368 
062023 
062879 
063134 
063,389 

10.063645 
06:3900 
0641.56 
064411 
0646(57 
064922 
065178 
06543:3 
065689 
06,5944 

10.066200 
0664rj5 
0(5(5711 
0(56967 
067222 
067478 
0677,34 
067990 
068245 
0(58501 

10.068757 
069013 
069269 
069,525 
069780 
070036 
070292 
070548 
070804 
071060 

10.071:316 
071573 
071829 
072085 
072341 
072597 
07285:3 
073110 
07:3:366 
07:3()22! 

10.073878 
0741:35 
074:591 


074648i^ 
074{)04i^' 
075160 1 J 
075417!;*' 
07567;'  ■* 


07,59:30 
076186 


Cotang.    PPl 


H«)() 


40" 


50'' 


SINES  AND  TANGENTS. 


51 » 


8*^360' 


88446« 
884572 
884677 
884783 
884889 
884994 
885100 
885205 
88.5311 
.885416 
88.5522 
88-5627 
885732 
88.5837 
885942 


886152 


886571 
886676 


887093 
887198 
887302 
887406 
9.887510 
887614 
887718 
887822 
887926 


888134 
888237 
888.341 
888444 

9.888548 
888651 
88875.- 
888858 
8889J1 
8.S90ol 
889168 
889271 
889374 
889477 

9.889579 
889382 
88978.J 
88J888 
889990 
89a093 


890400 
890503 


1.76 
1.76 
1.76 
1.76 
1.76, 
1.76 
1.76 
1.76 
1.76 
1.75 
1.75 
1.75 
1.75 
1.75 
1.75 
1.75 
1.75 
1.75 
1.75 
1.74 
1.74 
1.74 
1.74 
1.74 
1.74 
1.74 
1.74 
1.74 
1.74 
1.73 
1.73 
1.73 
1.73 
1.73 
1.73 
1.73 
1.73 
1.73 
1.73 
1.73 
1.72 
1.72 
1.72 
1.72 
1.72 
1.72 
1.72 
1.72 
1.72 
1.71 
1.71 
1.71 
1.71 
1.71 
1.71 
1.71 
1.71 
1.71 
1.71 


Cosine. 


Tang. 


10.076186 
076443 
076700 
0769-56 
077213 
077470 
077726 
077983 
078240 
078497 
078753 

10.079010 
079267 
079.524 
079781 
080038 
080295 
08a552 


081066 

081.323 

10.081.580 

0818:J7 


082352 


PPl' 


082866 
08:3123 
083381 
08:^638 
08;i896 

10.0841.53 
084} 10 
OS 4668 
0S4925 
0*5183 
085440 
085698 
0859.>() 
086213 
086471 

10.036729 
0SJ986 
087244 
087.')02 
0S7760 
088018 
0.S8275 
088533 
0S8791 
089049 

10.089;»7 
089565 
089823 
090082 
090340 
090-598 
090856 
091114 
091372 
091631 


PPl" 


4.27 

4.28 
4.28 
4.28 
1.28 
4.28 
4.28 
28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.28 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 


Cotang.    PIM 


M. 


9.89a503 
89060.5 
890707 


PI' 


890911 
891013 
891115 
891217 
891319 
891421 
89152:3 
9.891624 
891726 
891827 


892132 
89223:3 
892334 
8924:35 
892536 
9.892638 
892739 


893041 
893142 
893243 
893343 
893444 


9.893645 
89:3745 
89:5846 


894046 
894146 
894246 
894346 
894446 
894546 

9.894646 
894746 
894846 
894945 

,  895045 
895145 
895244 
895:343 
895443 
895542 

9.895641 
895741 
895840 
8959:39 
896038 
896137 
896236 
896335 
896433 
896532 


1.70 
1.70 
1.70 
1.70 
1.70 
1.70 
1.70 
1.70 
1.70 
1.70 
1.69 

i;69 
1.69 
1.69 
1.69 
1.69 
1.69 
1.69 
1.68 
1.68 
1.68 
1.68 
1.68 
1.68 
1.68 
1.68 
1.68 
1.68 
1-67 
1.67 
1.67 
1.67 
1.67 
1.67 
1.67 
1.67 
1.67 
1.67 
1.67 
1.66 
1.66 
1.66 
1.66 
1.66 
1.66 
1.66 
1.66 
1.66 
1.65 
1.65 
1.65 
l.a5 
1.6;5 
1.65 
1.65 
1.65 
1.65 
1.65 
1.65 


Tans. 


Cosine 


10.091631 
091889 
092147 
092406 
092664 
092923 
093181 
093440 
09:3698 
093957 
094215 

10.094474 
09473:3 
094992 
095250 
095509 
095768 
096027 
096286 
096544 


10.097062 
097321 
097580 
097!!M0 
098099 
098358 
098<il7 
098876 


10.099654 
099913 
100173 
100432 
100692 
100951 
101211 
101470 
101730 
101990 

10. 102249 
102509 
102769 
103029 
103288 
10:3548 
10:3808 
104068 
104328 
104588 

10.104848 
105108 
10-5368 
105628 
105889 
106149 
106409 
106669 
106930 
107190 


PPl" 

30 
4.31 
4.31 
4.31 
4.31 
4.31 
4.. 31 
4.31 
4.31 
4.31 
4.31 
4.31 
4.31 
4.31 
4.31 
4.31 
4.;31 
4.31 
4.31 
4.31 
4.32 
4.32 
4.32 
4.32 
4.32 
4.32 
4.32 
4.32 
4.32 
4.32 
4.32 
4.32 
4.32 
4.32 
4.32 
4.32 
4.3:3 

4.:33 

4.a3 
4.33 
4.33 
4.33 
4.33 
4.33 
4.33 
4.33 
4.33 
4.33 
4.33 
4.33 
4.  S3 
4.33 
4.33 
4.34 
4.34 
4.a4 
4.34 
4.34 
4.34 
4.34 


Cotang.    PPl 


39^ 


397 


3S" 


52  *> 


TABLE   IV.— LOGARITHMIC 


53« 


bine. 


37 


J.  896532 
896631 
896729 
896828 
896926 
897025 
897123 
897222 
897320 
897418 
897516 

9.897614 
897712 
897810 
897908 
898006 
898104 
898202 


l»l' 


898397 
898494 


898787 
898884 


899078 
899176 
899273 
899370 
899467 

9.899564 
899660 
899757 
899854 
899951 
900047 
900144 
900240 
900;«7 
900433 

9.900529 
900626 
900722 
900818 
900914 
901010 
901106 
901202 
901298 
901394 

9.901490 
90158.5 
901681 
901776 
901872 
901967 
902063 
902158 
902253 
902349 


Cosint 


1.64 
1.64 
1.64 
1.64 
1.64 
1.64 
1.64 
1.64 
1.64 
1.63 
1.63 
1.63 
1.63 
1.63 
1.63 
1,63 
1.63 
1.63 
1.63 
1.63 
1.63 
1.62 
1.62 
1.62 
1.62 
1.62 
1.62 
1.62 
1.62 
1.62 
1.62 
1.61 
1.61 
1.61 
1.61 
1.61 
1.61 
1.61 
1.61 
1.61 
1.60 
1.60 
1.60 
1.60 
1.60 
1.60 
1.60 
1.60 
1.60 
1.60 
1.60 
1.59 
1.59 
1.59 
1.59 
1.59 
1.59 
1.59 
1.59 
1.59 


'I'aiig.  IPPl"  M. 


10.107190 
107451 
107711 
107972 
108232 
108493 
108753 
109014 
109275 
1095:3.5 
109796 

10.110057 
110318 
110579 
110839 
111100 
111361 
111622 
111884 
112145 
112406 

10.112667 
112928 
113189 
113451 
113712 
113974 
1142.35 
114496 
114758 
115020 

10.115281 
115543 
115804 
116066 
116328 
116590 
116852 
117113 
117375 
117637 

10.117899 
118161 
118423 
118686 
118948 
119210 
119472 
1197a5 


120259 
10.120522 
1207R4 
121047 
121809 
121.372 
1218.35 
122097 
122360 
122623 
122886 


PPl"  Cotaiig.  PPl 


.36 


M. 


9.902349 
902444 
902;339 
902634 
902729 
902824 
902919 
903014 
903108 
903203 
903298 

9.903:592 
903487 
90;3581 
903676 
903770 
903864 


904053 
904147 
904241 

9.904335 
904429 
904523 
904617 
904711 
904804 
904898 
904992 
905085 
905179 

9.90.5272 
905366 
905459 
905552 
905645 
9067 
905832 
905925 
90C018 
906111 

9.906204 
906296 
906389 
906482 
906575 
90()667 
906760 
906852 
906945 
9070,37 

9.907129 
907222 
907314 
907406 
907498 
907590 
907682 
907774 
907866 
907958 


PPl" 

59 
1.59 
1.58 
l.,58 
1.58 
1.58 
1..58 
1..58 
1.58 
1.58 
1.58 
1.58 
1..57 
1..57 
1.57 
1.57 
1.57 
1.57 
1.57 
1.57 
1.57 
1.57 
1.57 
1.56 
1.56 
1.56 
1.56 
1.56 
1.56 
1.56 
1.56 
1.56 
1.55 
1.55 
1.55 
1.55 
1.55 
1.55 
L55 
1.55 
1.55 
1.55 
1.55 
1.55 
1.54 
1.54 
1.54 
1.54 
1.54 
1.54 
1.54 
1.54 
1.54 
1.53 
1.53 
1.53 
1.53 
1.53 
1.53 
1.53 


Tang. 


10.122886 
123148 
123411 
123674 
123937 
124200 
124463 
124727 
124990 
125253 
125510 

10.125780 
126043 
126306 
126570 
126833 
127097 
127360 
127624 
127888 
128151 

10.128415 
128679 
128943 
129207 
129471 
129735 
129999 
130263 
130527 
130791 

10.131055 
131320 
131584 
131848 
132113 
132377 
132642 
132906 
133171 
133436 

10.133700 
133965 
1M2S0 
134495 
134760 
1:35025 
135290 
ia5555 
135820 
136085 

10.13()a50 
1:36615 
136881 
137146 
137411 
137677 
137942 
138208 
138473 
1:38739 


CoKinf.  PPl"  (totalis 


38 


39 


3-8  • 


:-Ji)8 


PPl"  M. 

36* 


54' 


SINES  AND  TANGENTS. 


55' 


fSiiie 


IP  Pi" 


35" 


D.  9079.58  j 
9080i9  :*.„ 
908141 


9082;J8 
908;B:il 
908416 
908307 
908599 


908781 
908873 

9.9089134 
909055 
9(J9146 
9092.37 
909328 
909419 
909510 
909G01 
909091 
909782 

9.909873 
9099()3 
9100.34 
910144 
9102^5 
91032.5 
910415 
910.50() 
91059(5 


9.910776 
910.S66 
9109.56 
911046 
911136 
911226 
911315 
91140) 
911495 
9115H1 

9.911674 
911763 
91185:^ 
911942 
9120;^1 
912121 
912210 
912299 
912388 
912477 

9.912.566 
91265;5 
912744 
91283;^ 
912922 
91.3010 
913099 
913187 
913276 
918365 


1.53 
1..53 
1..53 
1.53 
1.52 
1..52 
1..52 
1.52 
1..52 
1.-52 
1..52 
1.-52 
1.52 
1..52 
1..52 
1..51 
1.-51 
1.51 
1..51 
1.-51 
1..51 
1..51 
1.51 
1.-51 
1.51 
1.50 
1.-50 
1..50 
1..50 
1.-50 
1.-50 
1.-50 
1.-50 
1..50 
1.-50 
1.-50 
1.50 
1.49 
1.49 
1.49 
1.49 
1.49 
1.49 
1.49 
1.49 
1.49 
1.49 
1.48 
1.48 
1.48 
1.48 
1.48 
1.48 
1.48 
1.48 
1.48 
1.48 
1.48 
1.47 


Tans.      PPr 


10.1-38739 
l,S90a5 
139270 
139=536 
139802 
1400(38 
140334 
140600 


141132 
141398 

10.141(j(>4 
141931 
142197 
142463 
142730 
142996 
143263 
14*529 
14:3796 
144032 

10.141329 
144-596 
144863 
145l:W 
145397 
145661 
145931 
146198 
146165 
148732 

10.146999 
147267 
147r>i4 
147801 
148069 
1483;J6 
148604 
148871 
149139 
149107 

10.149J75 
149943 
1.50210 
1-50478 
ir30746 
151014 
151283 
151551 
151819 
1-52087 

10.152356 
152624 
1-52892 
1-53161 
ir,3430 
1-53698 
1.53967 
154236 
154504 
154773 


4.43 
4.43 
4.43 
4.43 
4.43 
4.43 
4.43 
4.43 
4.43 
4.43 
4.44 
4.44 
4.44 
4.44 
4.44 
4.44 
4.44 
4.44 
4.44 
4.44 
4.45 
4.45 
4.45 
4.45 
4.45 
4.45 
4.45 
4.45 
4.45 
4.45 
4.45 
4.46 
4.46 
4.46 
4.46 
4.46 
4.46 
4.46 
4.46 
4.46 
4.46 
4.46 
4.47 
4.47 
4.47 
4.47 
4.47 
4.47 
4.47 
4.47 
4.47 
4.47 
4.47 
4.47 
4.48 
4.48 
4.48 
4.48 
4.48 
4.48 


Cosine.     PPl"    Cotaiigr.    PPl"    M 


M. 


feme. 


PPl' 


laiitr. 


9.913365 
9134.53 
913541 
913630 
913718 
913806 
913894 
913982 
914070 
914158 
914246 

9.914334 
914422 
914510 
914598 
914685 
914773 
914860 
914^8 
9150a5 
915123 

9.91,5210 
915297 
91.538.5 
915472 
91-55.59 
91.5646 
9157:« 
915820 
915907 
91-59{M 

9.916081 
916167 
916254 
916341 
91W27 
916514 
916600 
916687 
916773 
916859 

9.916946 
91703ii 
917118 
917201 
917290 
917376 
917462 
917548 
917634 
917719 

9.917805 
917891 
917976 
918062 
918147 
918233 
918318 
918404 
918489 
918574 


1.47 
1.47 
1.47 
1.47 
1.47 
1.47 
.47 
1.47 
1.47 
1.47 
1.47 
1.46 
1.46 
1.46 
1.46 
1.46 
1.46 
1.46 
1.46 
1.46 
1.45 
1.45 
1.45 
1.45 
1.45 
1.45 
1.45 
1.45 
1.45 
1.45 
1.45 
1.45 
1.45 
1.44 
1.44 
1.44 
1.44 
1.44 
1.44 
1.44 
1.44 
1.44 
1.44 
1.43 
1.43 
1.43 
1.43 
1.43 
1.43 
1.43 
1.43 
1.43 
1.43 
1.42 
1.42 
1.42 
1.42 
1.42 
1.42 
1.42 


10.154773 
155042 
155311 
155580 
155849 
156118 
156388 
156657 
156926 
157195 
157465 

10.157734 
158004 
158273 
158543 
158813 
159083 
159352 
159622 
159892 
160162 

10.160432 
160703 
160973 
161243 
161513 
1617i« 
1620;54 
162325 
162595 
162866 

10.163i;36 
16;i407 
163678 
163949 
164220 
164491 
164762 
165033 
165304 
165575 

10.165846 
166118 
166389 
166661 
166932 
167204 
16747 
16774' 
168019 
168291 

10.168563 
168835 
16910 
169371 
169651 
169923 
170195 
170468 
170740 
171013 


Cosine.  PPl"  Cot-nng.  PPl"  M 


PPl" 


49 
49 
50 
50 
50 
50 
50 
50 
50 
50 
50 
50 
51 
51 
,51 
,51 
,61 
,51 
,51 
,51 
,51 
,52 
,52 
,52 
,52 
,52 
,52 
,52 
,52 
,52 
,52 
.52 
.53 
.53 
.53 
53 
53 
53 
53 
53 
53 
53 
54 
54 
54 

54 


391) 


34' 


56' 


TABLE   IV.— LOGARITHMIC 


5^0 


Sine. 


9.918574 
918Ho9 
918745 
918830 
918915 
919000 
91908.5 
919169 
9192.54 
9193;?9 
919424 

9.919-508 
919593 
919677 
919762 
919846 
919931 
920015 
920099 
920184 
920268 

9.920352 
920436 
920520 
920004 
92068S 
920772 
920a56 
920939 
921023 
921107 

9.92119U 
921274 
921357 
921441 
921524 
921607 
921691 
921774 
9218.57 
921940 

9.922023 
922103 
922189 
922272 
9223.55 
922438 
922.520 
922603 


PP 


1.42 
1.42 
1.42 
1.42 
1.42 
1.41 
1.41 
1.41 
1.41 
1.41 
1.41 
1.41 
1.41 
1.41 
1.41 
1.41 
1.40 
1.40 
1.40 
1.40 
1.40 
1.40 
1.40 
1.40 
1.40 
1.40 
1.40 
1.40 
1.40 
1.-39 
1.39 
1.39 
1.39 
1.39 
1.39 
1.39 
1.39 
1.39 
1.39 
1.38 
1.38 
1.38 
1.38 
1.38 
1.38 
1.38 
1.38 
l.;38 
1.38 
1.38 
1.38 
1.37 
1.37 
1.37 
1.37 
1.37 
1.37 
1.37 
1..37 
1.37 


PPl" 


Taii«.  IP  PI" 


10.171013 
171285 
171558 
171830 
172103 
172376 
172649 
172922 
173195 
17M68 
173741 

10.174014 
174287 
174561 
174834 
175107 
17.5381 
17565.5 
175928 
176202 
176476 

10.176749 
177023 
177297 
177571 
177846 
178120 
178394 
178668 
178943 
179217 

10.179492 
179766 
180041 
180316 
180')90 
180865 
181140 
181415 
181690 
181965 

10.182241 
182-516 
182791 
183067 
ia3342 
18,3618 
183893 
184169 
184445 
184720 

10.184996 
185272 
185548 
1&5824 
186101 
186377 
186653 
186930 
187206 
187483 


PPl' 


M. 


9. 


923591 
92.3673 
9237.55 
92;S837 
923919 
924001 
9240&^ 
924164 
924246 
924.328 
924409 
924491 
924,572 
924654 
9247a5 
924816 
924897 
924979 
925060 
925141 
925222 
925,303 
925384 
925465 
925545 
925626 
92570' 
925788 
925868 
925949 
92€029 
.926110 
926190 
926270 
926,351 
926431 
926511 
926591 
926671 
926751 
926831 
.926911 
926991 
927071 
927151 
927231 
927310 
927390 
927470 
927549 
927G29 
.927708 
927787 
927867 
927M6 
928025 
928104 
928183 


928342 

928420 


PPl 


1.37 
1.37 
l.,37 
l.,36 
1.36 
1.36 
1.36 
1.36 
1.36 
1.36 
1.36 
1.36 
1.36 
1.36 
l.,35 
1.35 
1.35 

i.a5 
i.a5 

1.35 
l.,35 
1.35 
1.3-5 
1.35 
1.35 
1.34 
1.34 
1.34 
1.34 
1..34 
1,34 
1.34 
1.34 
1.34 
1.34 
l.,34 
1.33 
1.33 
1.33 
l.,33 
i.SS 
1.33 
1.33 
1.33 

i.;33 

1.33 
1.33 
1.33 
1.32 
1.32 
1.32 
1.82 
1.32 
1.32 
1.32 
1.32 
1.32 
1.32 
1.32 
1.32 


Taiie.  iPPl 


10.1874831 


1877591 


4.61 


188036 
188:^13 
188,5901 
188866! 
1891431 
1894201 


i4.61 
4.61 


189698 
189975 


4.61 
4.61 
4.62 
4.62 
4.62 
4.62 
^o:4.62 


190252 
10.190529^-^; 

191084  • 


191,362 
191639 
19191' 
192195 
192473 
192751 
193029 

10.193307 
19:3585 
193863 
194141 
194420 
194698 
194977 
19525,5 
1955^34 
195813 

10.196091 
196370 
196649 
196928 
197208 
197487 
197766 
198045 
198:325 
198604 

10.198884 
199164 
199443 
199723 
200003 
200283 
200563 
200843 
20112:3 
201404 

10.201684 
201964 
202245 
202526 
202806 
203087 
203368 
203649 


204211 


PP1"|  Cotaiie.    PPl" 


4.63 
4.63 
4.63 
4.63 
4.63 
4.63 
4.63 
4.63 
4.63 
4.64 
4.64 
4.64 
4.64 
4.64 
4.64 
4.65 
4.65 
4.65 
4.65 
4.65 
4.65 
4.65 
4.65 
4.65 
4.66 
4.66 
4.66 
4.66 
4.66 
4.66 
4.66 
4.67 
4.67 
4.67 
4.67 
4.67 
4.67 
4.67 
4.67 
4.68 
4.68 
4.68 
4.68 
4.68 
4.68 
4.68 
4.68 


3»' 


400 


3-^" 


58  « 


SINES  AND  TANGENTS. 


Sine. 

).  928420 
928499 
928578 
928657 
92S7m 
928815 
928893 
928972 
929050 
929129 
929207 

9.92928a 
92i);i61 
929142 
929521 
929599 
929677 
92975.5 
9298*^ 
929911 
929989 

9.9300t)7 
9;30145. 
93022;^ 
930300 
930378 
930456 
930.533 
930611 
930688 


I' PI" 


1.31 
1.31 
1.31 
1.31 
1.31 
1.31 
1.31 
1.31 
1.31 
1.31 
1..S1 
1..31 
1.30 
i.m 

\  i.m 
i.;jo 
i.;3o 

1..30 
l.:30 
1.30 
1.30 
1.30 
tl.30 
1.30 
1.29 
1.29 
1.29 
1.29 
1.29 

9307661 }-?2 
.9308431^*'^ 


930921 
9;W998 
931075 
9311.52 
931229 
931306 
931383 
931460 
931.537 
.931611 
931691 
931768 
931845 
931921 


1.29 
1.29 
1.29 
1.29 
1.29 
1.28 
1.28 
1.28 
1.28 
1.28 
1.28 
1.28 
1.28 
1.28 
1.28 


931998 i 
932075;,  r; 
9321.51  !-f! 
932218  •£; 

932.301  j,:' 

11.27 

1.27 
1.27 
1.27 
1.27 
1.27 
1.27 
1.27 
1.27 
1.27 


932457 
932533 
932609 
932685 
932762 
932838 
932914 
932990 
933066 


Cosinf 


10.204211 
204492 
204773 
20.5054 
205336 
20.5617 


206181 
206462 
206744 
207026 

10.207308 
207590 
207872 
2081.54 
208437 
208719 
209001 
209284 
209.566 
209849 

10.210132 
210115 
210698 
210981 
211264 
211.547 
21ia30 
212114 
212:^97 
212681 

10.212964 
213248 
2l;i532 
213816 
214100 
214384 
214668 
2149.52 
21.5236 
21.>521 

10.21.5805 
216090 
216.374 
216t).59 
210944 
217229 
217514 
217799 
218084 

2mm 

10.2186.54 
218940 
219225 
219511 
219797 
220082 
220368 
220654 
220940 
221226 


PP 


4.68 
4.69 
4.69 
4.69 
4.69 
4.69 
4.69 
4.69 
4.70 
4.70 
4.70 
4.70 
4.70 
4.70 
4.70 
4.70 
4.71 
4.71 
4.71 
4.71 
4.71 
4.71 
4.72 
4.72 
4.72 
4.72 
4.72 
4.72 
4.72 
4.72 
4.73 
4.73 
4.73 
4.73 
4.73 
4.73 
4.73 
4.73 
4.74 
4.74 
4.74 
4.74 
4.74 
1.75 
4.75 
4.75 
4.75 
4.75 
4.75 

He 

4.7o 

4.76 
4.76 
4.76 
4.76 
4.76 
4.76 
4.77 
4.77 
4.77 


PPl"    Cotang.    PPl 


).  9,3:^066 
9a3141 
93.3217 
933293 
9.3X369 
93:^45 
933520 
9.33596 
933671 
933747 
933822 


933973 
9^4048 
93412:5 
9^4199 
934274 
934349 
934424 
9344J)9 
9^4.574 

9.9»4649 
9^472;^ 
934798 
934873 
934948 
93.5022 
9a-;097 
935171 
935246 
9;i5;i20 

9.9a5395 
9;i>469 
9*5543 
9a5618 
9a5692 
9li5766 
9a5840 
9a5914 
935988 
9360«)2 

9.9:i6i:^i 
936210 
936284 
936a57 
93<)431 
936505 
936578 
93()652 
93(572.5 
936799 

9.936872 
936946 
937019 
937092 
937165 
9372;38 
937312 
93738.5 
937458 
937531 


Cosine.     PPl 


IM' 


59  » 

I  PPl";    31. 


1.26 
1.26 
1.26 
1.26 
1.26 
1.26 
1.26 
1.26 
1.26 
1.26 
1.26 
1.25 
1.25 
1.25 
1.2.5 
1.2.5 
1.25 
1.25 
1.25 
1.25 
1.2.5 
1.25 
1.2.5 
1.25 
1.24 
1.24 
1.24 
1.24 
1.24 
1.24 
1.24 
1.24 
1.24 
1.24 
1.24 
1.23 
1.2;} 
1.23 
1.23 
1.23 

1.2;? 

1.23 
1.23 
1.2;3 
1.23 
1.23 
1.23 
1.23 
1.22 
1.22 
1.22 
1.22 
1.22 
1.22 
1.22 
1.22 
1.22 
1.22 
1.22 
1.22 


10.221226 
221512 
221799 
222085 
222372 
222^58 
222945 
22.3232 
22a518 
223805 
224092 

10.224379 
2246(>7 
224954 
225241 
225529 
22.5816 
22t3104 
226392 
226679 
226967 

10.22725.5 
227543 
227832 
228120 
228408 


229274 
229.563 
229852 

10.230140 
2:J042J) 
230719 
231008 
231297 
231.586 
231876 
2.32166 
232455 
23274.5 

10.2;i'J0;iJ 

2^m2r, 

2;i3<il5 

2mo.'> 

234195 
2a4486 
2a4776 
2a5067 
235357 
235648 
10.235939 


2:^6521 
236812 
237103 
237394 
237686 
237977 


238561 


86 


Cotang.    PPl"    M 


31» 


Trii 


-34. 


401 


3a< 


60< 


TABLE   IV.— LOGARITHMIC 


61° 


).  937581 
937604 
937676 
937749 
937822 
93789.3 
937967 
938040 
938113 
938185 
9382.58 

).  938330 
938402 
938475 
938547 


938763 


).  939052 
939123 
939195 
939267 
939339 
939410 
939482 
939554 
939625 
939697 

).  939768 
939840 
939911 


940054 
940125 
940196 
940267 
940338 
940409 

),  940480 
940551 
940622 
940693 
940763 
940834 
9401)05 
940975 
941046 
941117 

9.941187 
941258 
941328 
941398 
941469 
941539 
941609 
941679 
941749 
941819 


PPl" 


1.21 
1.21 
1.21 
1.21 
1.21 
1.21 
1.21 
1.21 
1.21 
1.21 
1.21 
1.21 
1.20 
1.20 
1.20 
1.20 
1.20 
1.20 
1.20 
1.20 
1.20 
1.20 
1.20 
1.20 
1.19 
1.19 
1.19 
1.19 
1.19 
1.19 
1.19 
1.19 
1.19 
1.19 
1.19 
1.19 
1.18 
1.18 
1.18 
1.18 
1.18 
1.18 
1.18 
1.18 
1.18 
1.18 
1.18 
1.18 
1.18 
1.17 
1.17 
1.17 
1.17 
1.17 
1.17 
1.17 
1.17 
1.17 
1.17 
1.17 


rang.   PPl^M  3L 


10.238561 
238852 
239144 
239436 
239728 
240021 
240313 
240605 
240898 
241190 
24148;^ 

10.241776 
242069 
242362 
2426,55 
242948 
243241 
243535 
243828 
244122 
244415 

10.244709 
245003 
245297 
245591 
245885 
246180 
246474 
246769 
247063 
247358 

10.247^53 
247948 
248243 
248538 
248833 
249128 
249424 
249719 
250015 
250311 

10. 25060' 
250903 
251199 
251495 
251791 
25208 
252;:}84 
252681 
252977 
253274 

10.2r}a571 
2,53868 
2r;4165 
254462 
254760 
255057 
255355 
255652 
255950 
256248 


89 


M. 


0   9 

1 
2 
3 
4 
5 
6 
7 


34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 


,941819 
941889 
9419,59 
942029 
942099 
942169 
942239 
942308 
9423: 
942448 
942517 
.942587 
942656 
942726 
942795 
9428&4 
942934 
943003 
943072 
943141 
943210 
.943279 
943348 
943417 
943486 
943555 
943624 
943693 
943761 
943830 
943899 
1.943967 
944036 
944104 
9441 
944241 
944309 
944377 
944446 
944514 
944582 
1.944650 
944718 
944786 
944854 
944922 
944i)90 
94,5058 
945125 
945193 
D45261 
1.945328 
945396 
945464 
945531 
945598 
945666 
945733 
945800 
945868 


PPl" 


1.17 
1.16 
1.16 
1.16 
1.16 
1.16 
1.16 
1.16 
1.16 
1.16 
1.16 
1.16 
1.16 
1.16 
1.15 
1.15 
1.15 
1.15 
1.15 
1.15 
1.15 
1.15 
1.15 
1.15 
1.15 
1.15 
1.15 
1.14 
1.14 
1.14 
1.14 
1.14 
1.14 
1.14 
1.14 
1.14 
1.14 
1.14 
1.14 
1.14 
1-13 
1.13 
1.13 
1.13 
1.13 
1.13 
1.13 
1.13 
1.13 
1.13 
1.13 
1.13 
1.13 
1.12 
1.12 
1.12 
1.12 
1.12 
1.12 
1.12 


M.      Cosine.     PPl" I  Ootaiig.    PPl"    M.      M.  I  Cosine.    PPl"    Cotang.    PPl"    M 


Tang. 


10.256248 
256546 
256844 
257142 
2.57441 
257739 
258038 
258336 
258635 
258^34 
259233 

10.259,5.32 
259831 
260130 
260430 
260729 
261029 
261329 
261629 
261929 
262229 

10.262529 
262829 
263130 
263430 
26.3731 
264031 
264332 
26463:3 
264934 
265236 

10.265537 
265838 
266140 
266442 
266743 
267045 
267347 
267649 
267952 
268254 

10.208556 
268859 
269162 
269465 
26976' 
270071 
270374 
2706' 
270980 
271284 

10.271588 
271891 
272195 
272499 
272803 
273108 
273412 
273716 
274021 
274326 


•PI" 


4.97 
4.97 
4.97 
4.97 
4.97 
4.97 
4.98 
4.98 
4.98 
4.98 
4.98 
4.99 
4.99 
4.99 
4.99 
4.99 
4.99 
5.00 
5.00 
5.00 
5.00 
5.00 
,5.01 
,3.01 
5.01 
5.01 
5.01 
5.02 
5.02 
5.02 
5.02 
5.02 
5.02 
5.03 
5.03 
5.03 
5.03 
5.03 
5.04 
5.04 
5.04 
5.04 
5.04 
5.05 
5.05 
5.05 
5.05 
5.05 
5.06 
5.06 
5.06 
5.06 
5.06 
5.07 
5.07 
5.07 
5.07 
5.07 
5.08 
5.08 


29" 


402 


2S^ 


62" 


SLNES  AND  TANGENTS. 


63° 


iM. 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
2-5 
26 
27 
28 
29 
30 
31 
32 

m 

34 

aj 

36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
60 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 
M. 


Siiie.       PPl"      Tiiiiii.     iPP 


9. 9459^5 
940002 
946069 
946136 
946203 
946270 
946337 
946401 
946471 
916.538 
946604 

9.946671 
9467;38 
946804 
946871 

•  946937 
947001 
947070 
947136 
94720-3 
947269 

9.9473;ij 
947401 
947467 
9475;« 
947600 
94766.") 
947731 
947797 
94786:3 
947929 

9.94799') 
948060 
948126 
948192 
9482.^ 
94832:3 
948:388 
9484,54 
948.519 
948.384 

9.9486.50 
948715 
948780 
948845 


948975 
949040 
949105 
949170 
9492.^) 
).  949:300 
949:364 
949429 
949494 
949-558 
949623 
949688 
949752 
949816 
949881 


1.12 
1.12 
1.12 
1.12 
1.12 
1.11 
1.11 
1.11 
1.11 
1.11 
1.11 
1.11 
1.11 
1.11 
1.11 
i.U 
1.11 
1.11 
1.10 
1.10 
1.10 
1.10 
1.10 
1.10 
1.10 
1.10 
1.10 
1.10 
1.10 
1.10 
1.10 
1.09 
1.09 
1.09 
1.09 
1.09 
1.09 
1.09 
1.09 
1.09 
1.09 
1.09 
1.09 
1.08 
1.08 
1.08 
1.08 
1.08 
1.08 
1.08 
1.08 
1.08 
1.08 
1.08 
1.08 
1.08 
1.08 
1.07 
1.07 
1.07 


10.274326; 
274&30i^^^ 
27493.5|'^-"^ 

iv- oinK)«Oo 

27.O240; 
275.546  ?-X„ 


27,58.51 

2761-56 

2764621"; 

276768 

277073 

277379 

10.27768.5 
277991 
278298 
278604 
278911 
279217 
279.524 
2798:31 
2801:38 
280445 

10.2807,52 
2810(30 
281:367 


5.( 


5.09 
5.10 
5.10 
5.10 
.5.10 
5.10 
,5.11 
.5.11 
5.11 
.5.11 
.5.11 
.5.12 
,5.12 
.5.12 
.5.12 
5.12 
13 


281675:  ,o 
2819*3  '^ 


282291 
282599 
282907 
2*3215 
28:3.523 

10.2838.32 
284140 
284449 
284758 
28,5067 
28,5376 
28,5686 
28.5995 
286:304 
286614 

10.286924 
287234 
287544 
2878.54 
288164 
288475 
288785 
289096 
28940' 
289718 

10.290029 
290:540 
2906.51 
2909()3 
291274 
291586 
291898 
292210 
292.522 
292834 


iiie.     PPl"!  (' 


.13 

.13 
5.13 
5.14 
5.14 
j.l4 
5.14 
5.14 
5.15 
5.15 
5.15 
5.15 

.16 
.5.16 
5.16 
5.16 
5.16 
.5.17 
5.17 
5.17 
5.17 
5.18 
5.18 
5.18 
5.18 
.5.18 
5.19 
5.19 
5.19 
5.19 
5.19 
5.20 
,5.20 
.5.20 
5.20 

ppT" 


M. 

M. 

60 

0 

59 

1 

58 

2 

57 

3 

56 

4 

55 

5 

54 

6 

53 

7 

52 

8 

51 

9 

50 

10 

49 

11 

48 

12 

47 

13 

46 

14 

45 

15 

44 

16 

43 

17 

42 

18 

41 

19 

40 

20 

39 

21 

38 

22 

37 

23 

36 

24 

a5 

25 

34 

26 

33 

27 

32 

28 

31 

29 

30 

30 

29 

31 

28 

32 

27 

a3 

26 

34 

25 

35 

24 

36 

23 

37 

22 

38 

21 

39 

20 

40 

19 

41 

18 

42 

17 

43 

16 

44 

15 

45 

14 

46 

13 

47 

12 

48 

11 

49 

10 

50 

9 

51 

8 

52 

7 

53 

6 

54 

5 

55 

4 

56 

3 

57 

2 

58 

1 

59 

0 

117 

60 

M. 

iPPl"      Tanir.      PPl" 


9.9498811   , 
949945:!'"' 
950010,  "' 
9.50074  }•"' 

a5oi3s:J-"' 

950202 1 ;•"' 

950266  \'^ 

950330  ,  ^' 

950:394  ;•"' 

9504.58  !'": 

950522 1 !•"' 

9.950586!,  "^ 
l.Ob 

1.06 

1.06 

1.06 

1.06 

1.06 

1.06 

1.06 

.06 

.06 

1.06 

1.06 

1.05 

1.05 

1.05 

1.05 

1.05 

1.05 

1.05 

1.05 

1.05 

1.05 

1.05 

1.05 

1.05 

1.04 

1.04 

1.04 

1.04 

1.04 

1.04 

1.04 

1.04 

1.04 

1.04 

1.04 

1.04 

1.04 

1.03 

1.03 

1.03 

1.03 

1.03 

1.03 

1.03 

1.03 

1.03 

1.03 

1.03 


950650 
950714 
9.50778 
9.50841 
950905 
950968 
95103'2 
951096 
951159 

9.951222 
951286 
951:349 
951412 
951476 
951539 
951602 
9516(55 
951728 
951791 

9.951854 
951917 
9,51980 
952043 
952106 
952168 
952231 
952294 
952356 
952419 

9.952481 
952544 
952606 
952669 
952731 
952793 
952855 
952918 
952980 
953042 

9.953104 
953166 
9,53228 
953290 
9,5:3,3.52 
953413 
953475 
953,5.37 
95:3599 
9.53660 

Cosine. 


PI'l" 


10.2928:34 
293146 
293459 
293772 
294081 
2943.97 
294710 
2950241 


5.21 
5.21 
5.21 
5.21 
.5.21 
5.22 
-5,22 


295:337  r^'l 

295650 '^ff 

295964!^-g 

10.296278  "^'^ 


296.591 
296905 
297219 
297534 
297848 
298163 


5.23 
5.23 
5.23 
5.24 
5.24 
5.24 
5.24 

298792  ?-^f 
299107  i^-f"? 
10.29W22|:?-r? 
2997371^-2^ 
3000531--^^ 
300:UJ8  ^'^^ 


300<J84 
300999 
301315 
301631 
301947 
302264 
10.302580 
302897 
303213 
30a530l?'f 
303847 
304164 
304482 
301799 
305117 


5.26 
5.26 
5.26 
5.27 
0.27 
5.27 
5.27 
5.28 
5.28 


5.28 
5.29 
5.29 
5.29 
5.29 


o0o4t>4  -  .,« 
10.305752  ^-^^ 


306070 
306:388 
306707 
307025 
S07344 
307662 
307981 
308300 
308619 


309258 
309577 


310217 
310537 
310857 
311177 
311498 
311818 


5.30 
5.30 
5.30 
5.31 
5.31 
5.31 
5.31 
5.31 
5.32 
5.32 
5.32 
5.33 
5.33 
5.33 
5.33 
5.33 
5.34 
5.34 
5.34 


403 


20  « 


64» 


TABLE   IV.— LOGARITHMIC 


65° 


40 
41 
42 
43 
-44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.9.53(560 
953722 
95878;^ 
9.5;«4r 
95390<i 
9rj39()8 
9.54029 
9,54090 
954152 
9-54213 
9.54274 

9.954335 
954390 
9544,57 
954518 
954579 
954640 
9.51701 
954762 
95182;^ 
95188;^ 

9.954944 
9550a5 
955065 
955126 
95.5186 
9,55247 
955307 
8.55368 
955428 
9,55488 

9.9,55,548 
9,55(509 
9-55669 
9.5;5729 
9,5.5789 
95.5849 
9.55909 
9.55969 
956029 
9560.S9 

9.956148 
956208 
956268 
9,56,327 
95(5387 
9,5(5447 
9,56.50(5 
956.5(56 
956625 
9.56684 

9.9.56744 
9568(KJ 
956862 
956921 
956981 
9.57040 
957099 
957158 
957217 
957276 


Cosin* 


I'J>1"   Tail!?. 


1.03 
1.02 
1.02 
1.02 
1.02 
1.02 
1.02 
1.02 
1.02 
1.02 
1.02 
1.02 
1.02 
1.02 
1.01 
1.01 
1.01 
1.01 
1.01 
1.01 
1.01 
1.01 
1.01 
1.01 
1.01 
1.01 
1.01 
1.01 
1.01 
1.00 
1.00 
1.00 
1.00 
1. 00 
1.00 
1.00 
1.00 
1.00 
1.00 
1.00 
1.00 
1.00 
.99 
.99 
.99 
.99 
.99 
.99 
.99 
.99 
.99 
.99 
.99 
.99 
.98 
.98 
.98 
.98 


10.311818 
312139 
312460 
312781 
313102 
313423 
31,3745 
314066 
314,388 
314710 
31.5032 

10.31.5354 
315676 
31599f) 
316321 
316644 
316967 
317290 
317613 
317937 
318260 

10.318584 
318908 
319232 
319556 
319880 
320205 
320,529 
320854 
321179 
321.504 

10.321829 
322154 
322480 
322806 
32^131 
323457 
323783 
324110 
324436 
324763 

10.325089 
325416 
325743 
326071 
326398 
326726 
327053 
327381 
327709 
328037 

10.328365 
328694 
32902;^ 
329;i51 
329680 
3:30009 
a30339 
330668 
3^30998 
31^327 


PPI'M  (^ot 


I' I' 


5.34 
.5.-3^5 
5.35 
5.35 
5,35 
5.36 
5.36 
.5.36 
5.36 
5.37 
5.37 
5.37 
.5.37 
5.38 
5..38 
5.38 
5.38 
5.39 
5.39 
5.39 
5.39 
5.40 
5.40 
5.40 
5.40 
5.41 
-5.41 
-5.41 
5.41 
5.42 
5.42 
,5.42 
5.42 
5.43 
5.43 
.5.43 
5.43 
5.44 
.5.44 
,5.44 
5.44 
5.45 
5.45 
5.45 
5.46 
5.46 
5.46 
5.46 
5.47 
5.47 
5.47 
5.47 
5.48 
5.48 
5.48 
5.48 
5.49 
5.49 
5.49 
5.50 


PP1' 


Hinc 


).  9.57276 
95733.5 
957393 
9.57452 
9,57511 
957.570 
957628 
957687 
957746 
957804 
957863 

).  957921 
957979 
958038 
958096 
958154 
958213 
958271 
958329 
958387 
958445 

).958503 
958561 
958619 
958677 
958734 
958792 


958965 
959023 

.959080 
959138 
959195 
9592.53 
959310 
959368 
9.59425 
959482 
959,539 
959-596 

.9596.54 
959711 
959768 
959825 
959882 
959938 
959995 
960052 
9t)0109 
960165 

.960222 
960279 
9603a5 
9(i()392 
960448 
96a50.5 
9(50561 
960618 
960674 
960730 


PP 


PPl" 


Tang.      PPl"    M. 


10. 


).  331327 
3316.57  ^''c: 
331987  'I'l^ 
332318  ^-.y 
a32648  "^•?{ 
332979  J'^l 
33;3309  .'f 
333640!?"^; 
33.3971 1^-^^ 
3.34302^-J^ 
334634?*?^ 

).331965?*^ 
3352971^-^; 
335629  ^-e^ 
335961  l'^ 

as6293  rrj 

33662.5  :^-3 

336958  :?•:! 

337291  ^•?! 
337624  ^f 

337ft57  r*;^ 

).338290  ?*^ 
338623  iT^ 

339290?'^ 

5.57 
5.57 
5.57 
5.58 
5.58 
5.58 
5.58 
5..59 
5.59 
5.59 
5.59 
5.60 
5.60 
.5.60 
5.61 
5.61 
5.61 
5.61 
5.62 
5.62 
5.62 
5.63 
5.63 
5.63 
5.63 
.5.64 
5.64 
5.64 
5.65 
5.65 
5.65 


339958 
ai0292 
340627 
340961 
341296 
1.341631 


..66 


404 


«4^ 


I  06« 


SINKS  AND  TANGENTS. 


67  0 


23^ 


Siup. 


>.  960730 
960786 
960843 
960899 
96095.5 
961011 
961067 
961123 
961179 
961235 
961290 

).  961346 
961402 
9614,58 
961513 
961.5t39 
961624 


9617;i5 
961791 
901816 

9.961902 
9619.57 
962012 
962067 
962123 
962178 
9622:i3 
962288 
962343 
902308 

9.9624-')3 
9<52508 
962562 
962617 
962672 
962727 
962781 
9628;36 
962890 
962945 

9.902999 
963054 
963108 
96:3163 
963217 
963271 
963325 
96;«79 
96:3434 
963488 

9.963:542 
96:3.593 
96:3a50 
96:5701 
90375 
963811 
96386,5 
963919 
963972 
964026 


PPI' 


.93 
.93 
.93 
.93 
.93 
.93 
.93 
.93 
.92 
.92 
.92 
.92 
.92 
.92 
.92 
.92 
.92 
.92 
.92 
.92 
.92 
.92 
.91 
.91 
.91 
.91 
.91 
.91 
.91 
.91 
.91 
.91 
.91 
.91 
.91 
.90 
.90 
.90 
.90 
.90 
.90 
.90 
.90 
.90 
.90 
.90 
.90 
.90 
.90 


I'Fl 


l«f;ii!|5.67 


a52097 
352438 
3.52778 
a53119 
3-53460 
a5;3801 
a54143 
354484 
a54826 

10.,a>5168 
a>5.510 
3558.52 
a56194 
3.5a5:37 
,356880 
a57223 
a57566 
a57909 
3.582.5:3 

10.358.596 
a58940 
359284 
a59629 
a59973 
.360318 
36066:3 
361008 
361:3.53 
361698 

10.362014 
362:389 
3627.a5 
363081 
S(m2S 
363774 
364121 
364468 
364815 
365162 

10.365510 
36.5857 
366205 
.366.553 
366901 
3672.50 
367598 
367947 
368296 
368645 

10.:368995 
309344 
369694 
370044 
370394 
370745 
371095 
371446 
371797 
372148 


Cosine.     PPl' 


5.67 
5.68 
5.68 
.5.68 
5.69 
5.69 

5.69 
5.70 
5.70 
5.70 
5.71 
5.71 
5.71 
5.72 
5.72 
5.72 
5.72 
5.73 
5.73 
5.73 
5.74 
5.74 
5.74 
5.75 
5.75 
5.75 
5.76 
5.76 
5.76 
5.77 
,5.77 
5.77 
5.77 
.5.78 
5.78 
5.78 
5.79 
5.79 
5.79 
5.80 
5.80 
5.80 
,5.81 
5.81 
5.81 
5.82 
5.82 
5.82 
5.83 
5.83 
,5.83 
5.83 
.5.84 
5.84 
5.85 
5.85 
.5.85 


Cotang. 


M. 


PPl" 


9.964026 
964080 
964133 
9^4187 
964240 
964294 
964347 
964400 
964454 
964.507 
964560 

9.964613 
964666 
964720 
964773 
9W826 
964879 
9649:31 
9<>4984 
965037 
9ft5090 

9.965143 
96;5ia5 
965248 
965:301 
9ft5a53 
96.540(3 
96;54.58 
96i5511 
96.5.503 
965615 

9.965668 
965720 
965' 
965824 
965876 
965929 
965981 
9660:33 
966085 
966136 

9.966188 
966240 
966292 
966344 
906:395 
9()6447 
966499 
9665;50 
966602 
966653 

9.9667ft5 
966756 
966808 
966859 
966910 
966961 
967013 
9670(34 
967115 
9671(30 


89 


:  10. 372148 
372499 
372851 
373203 
37a55,5 
373907 
374259 
374612 
374964 
37,5317 
37.5670 

10.370024 
.376377 
376731 
37708.5 
3774:39 
377793 
378148 
378.503 
378858 
379213 

10.379.568 
379924 
3802S0 
3806.36 
380992 
.381.348 
3817ft5 
382061 
382418 
382776 

10..38;31.3;3 
38:3491 
38;3849 
38420' 
aS4.56;5 
38492:3 
38.5282 
385(>11 
386000 
386:3,59 

10..386719 
387079 
387439 
aS7799 
3881.59 
aS8520 
388880 
389241 
389603 


10.390320 
390088 
391050 
391412 
391775 
392137 
392.500 


393227 
393590 


5.85 
5.86 
5.86 
5.86 
5.87 
5.87 
5.87 
,5.88 
5.88 
5.88 
5.89 
5.89 
5.89 
5.90 
,5.90 
.5.90 
5.91 
5.91 
5.92 
5.92 
.5.92 
5.93 
5.93 
5.93 
5.94 
5.94 
,5.94 
5.95 
,5.95 
5.95 


5.97 
5.97 
5.97 
5.98 
5.98 
5.98 


0.00 
6.00 
6.00 
6.01 
6.01 
6.01 
0.02 
0.02 
6.02 
6.03 
6.03 
6.03 
6.04 
6.04 
6.04 
6.05 
6.05 
6.06 
6.06 


Cosino.  PPl" I  (?otang. 


PPl' 


405 


22' 


68° 


TABLE   IV.— LOGARITHMIC 


69  < 


Sino. 


9.967166 
967217 
967268 
967319 
967370 
967421 
967471 
967522 
967573 
967624 
967674 

9.967725 
967775 
967826 
967876 
967927 
967977 
968027 
968078 
968128 
968178 

9.968228 
968278 
968;329 
968379 
968429 
9684 
968528 
968578 
968628 
968678 

9.968728 
968777 
968827 
968877 
968926 
968976 
969025 
969075 
969124 
969173 

9.969223 
969272 
969321 
969370 
969420 


969518 
96956' 


969665 

).  969714 

969762 

969811 


970006 
9700.55 
970103 
970152 


10.393.390 
393954 
394318 
394683 
39504 
395412 
395777 
396142 
396507 
396873 
397239 


10.397605  ^'^^ 


Taiu 


I'l' 


6.06 
6.07 
6.07 

:::6.o7 

6.08 
6.08 
6.09 
6.09 
5.09 
5.10 


397971 
398337 
398704 
399071 


400173 
400541 
400909 

10.401278 
401646 
402015 
402384 
402753 
403122 
403492 
403862 
404232 
404602 

10.404973 
405344 
405715 
406086 
406458 
406829 
407201 
407574 
407946 


408319  ^E 


10.408692 
409065 
409438 
409812 


410186 
410560  ^"^ 


6.10 
6.11 
6.11 
6.11 
6.12 
6.12 
6.13 
6.13 
6.13 
6.14 
6.14 
6.15 
6.15 
6.15 
6.16 
6.16 
6.16 
6.17 
6.17 
6.18 
6.18 
6.18 
6.18 
6.19 
6.19 
6.20 
6.20 
6.21 


6.22 
6.22 
6.22 
6.23 
6.23 


410934 
411309 
411684 
412059 
10.412434 
412810 
41318.5 
4ia561 


414314 
414691 
415068 
415445 
415823 


6.24 
6.24 
6.2.5 
6.25 
6.25 
6.26 
6.26 
6.27 
6.27 
6.27 
6.28 
6.28 
6.29 
6.29 


M. 


ill.'.   PPi 


9.970152 
970200 
970249 
970297 
970345 
970394 
970442 
970490 
970,538 
970586 
■9706a5 

9.970683 
970731 
970779 
970827 
970874 
970922 
970970 
971018 
971066 
971113 

9.971161 
971208 
971256 
971303 
971351 
971398 
971446 
971493 
971540 
971588 

9.971635 
971682 
971729 
97r 
971823 
971870 
971917 
971964 
972011 
972058 

9.972105 
972151 
972198 
972245 
972291 
972338 
97238.5 
972431 
972478 
972524 

9.972570 
972617 
972663 
972709 
972755 
972802 
972848 
972894 
972940 
972980 


M.   Cosine.  PPl"  Cotang.  I  PPl"  31.   M.  |  Cosine.  PPl"  CotHUg.  |PPl"  M 


.79 


1  .7: 


rang.   PPl"  31. 


10.415823 
416200 
416578 


417335 
417714 


418472 
418851 
419231 
419611 

10.419991 
420371 
420752 
421133 
421514 
421896 
42227' 
422659 
423041 
423424 

10.423807 
424190 
424573 
424956 
425340 
425724 
426108 
426493 
426877 
427262 

10.427648 
428033 
428419 
428805 
429191 
429578 
429965 
430;%2 
430739 
431127 

10.431514 
4319021 
432291 
432680 
433068 
433458 
433847 
434237 
434627 
435017 

10.435407 
435798 
436189 
436581 
436972 
437364 
437756 
438149 
438541 
438934 


29 
6.30 
6.30 
6.31 
6.31 
6.32 
6.32 
6.32 
6.33 
6.a3 
6.34 
6.34 
6.34 


6.35 
6.36 
6., 36 
6.36 
6., 37 
6.37 
6.38 
6.38 
6.39 
6.39 
6.39 
6.40 
6.40 
6.41 
6.41 
6.42 
6.42 
6.42 
6.43 
6.43 
6.44 
6.44 
6.45 
16.45 
6.45 
6.46 
6.46 
16.47 
6.47 
6.48 
6.48 
6.49 
6.49 
6.49 
6.50 
6.50 
6.51 
6.51 
6.52 
6.52 
6.53 
6.53 
6.53 
6.54 
6.54 
6.55 


21" 


406 


20' 


70° 


SINES  AND  TANGENTS. 


nr 


M. 


Si  lie 


9.972980 
97;«)82 
973078 
973124 
973169 
973215 
973261 
973:307 
973;352 
973398 
973144 

9.973489 
97.35;35 
973-580 
97362.5 
973671 
973716 
973761 
973807 
973852 
97;3897 

9.973942 
973987 
974032 
974077 
974122 
974167 
974212 
974257 
974302 
974347 

9.974391 
974436 
974481 
97452.5 
974.570 
974614 
9746.59 
974703 
974748 
974792 

9.974836 
974880 
97492.5 
974969 
97,5013 
97.5057 
975101 
975145 
975189 
9752;« 
9.975277 
975321 
975365 
975J08 
975452 
97M96 
975539 
975583 
975627 
975670 


M.   Cosine.  PPl" 


.77 


.76 
.76 
.76 
.76 
.75 
.75 
.75 
.75 
.75 
.75 
.75 
.75 
.75 
.75 
.75 
.75 
.75 
.75 
.75 
.74 
.74 
.74 
.74 
.74 
.74 
.74 
.74 
.74 
.74 
.74 
.74 
.74 
.74 
.73 
.73 
.73 
.73 
.73 
.73 
.73 
.73 
.73 
.73 
.73 
.73 
.73 
.73 
.73 
.73 


Tang.  PPl"  M. 
10.4.38934 
439327 
439721 
440115 
440509 
440903 
441297 
441692 
442087 
442483 
442879 
10.443275 
443671 
444067 
444464 
444861 
44.52.59 
44.5656 
4460.51 
4464.52 
4468.51 
10.4472.50 
447649 
448018 
448418 
418847 
449218 
449648 
4;50049 
4.504.50 
4.508,51 
10.4512.53 
4516.55 
4.520.57 
4.52460 
4.52862 
4,53265 
4.53669 
454072 
454476 
4.54881 
10.45528.5 

4,5.5690 

45(5095 

4,56501 

45690() 

457312 

4,57719 

4,5812.5 

458,5:32 

4.58939 
10.4.59347 

459755 

460163 

460.57 

460980 

461389 

461798 

462208 

462618 

463028 


6.55 
6.,56 
6.56 
6.-57 
6.57 
..58 
.58 
6.59 
6.59 
.59 
.60 
6.60 
6.61 
6.61 
6.62 
6.62 
6.63 
6.63 
6.64 
6.64 
6.6.5 
6.65 
6.6,5 
6.66 
6.66 
6.67 
6.67 
6.68 
6.68 
6.69 
6.69 
6.70 
0.70 
6.71 
6.71 
6.72 
6.72 
6.73 
.73 
6.74 
6.74 
6.75 
6.75 
6.76 
6.76 
6.77 
6.77 
6.78 
6.78 
6.79 
6.79 
6.80 
6.80 
6.81 
6.81 
6.82 
6.82 
6.83 
6.83 
6.84 


5 
4 
3 
2 
1 
0 
Cotang.  PPl"  M. 


PPl' 


9.975670 
975714 
9757,57 
975800 
975844 
975887 
975930 
975974 
976017 
976060 
976103 

9.976146 
976189 
9762:32 
976275 
976318 
976361 
97{>104 
976446 
976489 
976532 

9.976.574 
976617 
9766(i0 
976702 
976745 
976787 
9768:30 
976872 
976914 
976957 

9.976999 
977041 
977083 
97712.5 
977167 
977209 
977251 
9772i<3 
9773:35 
977377 

9.977419 
977461 
977503 
977544 
977586 
977628 
977669 
977711 
977752 
977794 

9.977835 
977877 
977918 
977959 
978001 
978042 
978083 
978124 
978165 
978206 


Cosine 


.72 

.72 
.72 
.72 
.72 
.72 
.72 
.72 
.72 
.72 
.72 
.72 
.72 
.71 
.71 
.71 
.71 
.71 
.71 
.71 
.71 
.71 
.71 
.71 
.71 
.71 

.71 
.70 
.70 
.70 
.70 
.70 
.70 
.70 
.70 
.70 
.70 
.70 
.70 
.70 
.70 
.70 
.70 


.69 


.69 
.69 


.68 
.68 


Inns.      PPl"  31. 


10.463028 
46:3439 
463850 
464261 
464672 
465084 
465496 
465908! 
466321 
466734 
467147 

10.467561 
467975 
468:389 
468804 
469219 
4696:34 
47'X)49 
470465 
470881 
471298 

10.471715 
4721:32 
472,519 
472967 
473385 
473803 
474222 
474641 
475060 
475480 

10.475900 
476320 
476741 
477162 
47758^3 
4780a5 
478427 
478849 
479272 
479695 

10.480118 
48a542 
48096(5 
481:390 
481814 
4822.39 
48266.5 
483090 
483516 
483943 

10.484369 
484796 
485223 
485651 
486079 
486507 


PPl" 


487365 
487794 
488224 


6.84 
6.85 
6.85 
6.86 
6.86 
6.87 
6.87 
6.88 
6.88 


.90 
6.90 
6.91 
6.91 
6.92 
6.93 

.93 
6.93 
6.94 
6.95 
6.a5 
6.96 
6.96 
97 
6.97 
6.98 
6.98 

.99 
6.99 
7.00 
7.01 
7.01 
7.02 
7.02 
7.03 
7.03 
7.03 
7.04 
7.05 
7.05 
7.06 
7.06 
7.07 
7.08 
7.08 
7.09 
7.09 
7.10 
7.10 
7.11 
7.12 
7.12 
7.13 
7.13 
7.14 
7.14 
7.15 
7.16 
7.16 


19^ 


407 


Cotang.    PPl"!  M. 
__ 


12° 


TABLE   IV.— LOGARITHMIC 


73" 


.978206 
978247 
978288 
978329 
978370 
978411 
978452 
978493 
978533 
978.574 
978615 
.978655 
978696 
978737 
978777 
978817 
978858 
978898 
978939 
978979 
979019 
,979059 
979100 
979140 
979180 
979220 
979260 
979300 
979340 
979380 
979420 
,979459 
979499 
979539 
979579 
979618 
979658 
979697 
979737 
979776 
979816 
,979&55 
979895 
979934 
979973 
980012 
980052 
980091 
980130 
980169 
980208 
,980247 
980286 
98032;5 
980364 
980403 
980442 


980519 
980.558 
980596 


M.   OosiiiH.  PPi" 


Thus.  riM"  M^ 
60 
59 
58 
57 
56 
5.5 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 


10.488224 
4886.54 
4«9081 
489515 
489946 
490378 
490809 
491241 
491674 
492107 
492540 

10.492973 
493407 
49.3841 
494276 
494711 
495146 
495.582 
496018 
4964,54 
496891 

10.497328 
4977a5 
498203 
498641 
499080 
499519 
499958 
500397 
500837 
501278 

10.501718 
,502159 
502601 
503043 
503485 
503927 
504370 
504814 
5052,57 
50.5701 

10.506146 
506590 
507035 
507481 
507927 
508373 
508820 
509267 
509714 
.510162 

10.510610 
5110;)9 
,511,508 
5119.'.7 
512407 
5128.57 
5i;«07 
5137.58 
514209 
514661 


Cotiuiir. 


Pl'l' 


&iti( 


).  980596 
98063.5 
9806' 
980712 
9807.50 
980789 
980827 
980866 
980904 
980942 


9.981019 

98105 

981095 

981133 

981171 

981209 

981247 

981285 

981323 

981361 
9.981399 

981436 

981474 

981512 

981549 

981587 

981625 

981662 

981700 

981737 
9.981774 

981812 

981849 

981886 

981924 

981961 

981998 

982035 

982072 

982109 
9.982146 

98218^3 

982220 

982257 

982294 

982a31 

982367 

982404 

982441 

982477 
9.982514 

982.551 

982.587 

982624 

982()(i(l 

982{)9<) 

982733 

9827(i9 

982805 

982842 

Cosine.    PPI 


'Pi" 


Tang. 


10.514661 
515113 
51,556.5 
516018 
516471 
516925 
517379 
517833 
518288 
618743 
519199 

10.519655 
520111 
520568 
521025 
521483 
521941 
522399 
5228.58 
523317 
523777 

10.524237 
524697 
525158 
525619 
526081 
526543 
527005 
527468 
627931 
628395 

10.528859 
529324 
529789 
630254 
530720 
531186 
531653 
532120 
532587 
533065 

10.53352,3 
533992 
534461 
634931 
635401 
636872 
636;S42 
636814 
637285 
637758 

10.5,38230 
a38703 
639177 
639651 
640125 
510600 
541075 
541551 
642027 
542504 

Cotang. 


11< 


408 


PPl'^    M. 

16^ 


74' 


SINES  AND  TANGENTS. 


750 


Si  I 


9.982842 
982878 
982914 
9829.30 


983022 
983038 
983094 
983130 
983166 
983202 

.983238 
983273 
983309 
983345 
983381 
983416 
983452 
983487 
983523 
9835^38 

.983594 
983629 
983664 
983700 
9837a3 
983770 


983840 
983875 


984015 
9840.30 
98408.3 
984120 
984155 
984190 
984224 
9842.39 

9.981294 
984328 
984363 
984397 
984432 
984466 
984500 
984535 
981569 
984603 

9.984638 
984672 
984706 
984740 
984774 
984808 
984842 
984876 
984910 
984944 


Cosii 


..59 


.59 


Tang. 


10.542.504 
542981 
5431.58 
543936 
544414 
544893 
545372 
545852 
546332 
546813 
547294 

10.547775 
548257 
548740 
549223 
549706 
550190 
550674 
551159 
551644 
552130 

10.552616 
533102 
653589 
554077 
554.365 
535053 
53.3.342 
536032 
556521 
557012 

10.557503 
557994 
558486 
558978 
559471 
559964 
560457 
560952 
561446 
561941 

10.562437 


563430 
563927 
564424 
561922 
56.3421 
565920 
566420 
566920 
,567420 
567921 
568423 
568925 
569427 
5699,30 
570434 
5709;« 
571442 
571948 


15 


Trig.— 35. 


ppi' 


7.95 
7.96 
7.96 
7.97 
7.98 
7.99 
7.99 
8.00 
8.01 
8.02 
8.02 
8.03 
8.04 
8.05 
8.08 
8.06 
8.07 
8.08 
8.09 
8.09 
8.10 
8.11 
8.12 
8.12 
8.13 
8.14 
8.15 
8.16 
8.16 
8.17 
.18 
8.19 
8.19 
8.20 
8.21 
8.22 
8.23 
8.23 
8.24 
8.25 
8.26 
8.27 
8.28 
8.28 
8.29 
8.30 
8.31 
8.32 
8.32 
8.3:3 
8.34 
8.35 
8.36 
8.37 
8.38 
8.38 
8.39 
8.40 
8.41 
8.42 


M. 


I'T'l' 


Sine. 


iPPl" 


.9849441 
984978 
985011 
985045 
985079 
985113 
985146 
985180 
985213 
985247 


9.985314 
985^47 
985381 
985414 
985447 
985480 
985514 
983547 
985580 
985613 

9.985646 
985679 
985712 
985745 
985778 
985811 


985876 
985909 
985942 

9.98.3974 
986007 
986039 
986072 
986104 
98(5137 
986169 
986202 
986234 
986266 

9.986299 
986331 
986363 


986427 
986459 
986491 
986523 
986555 
986587 
9.986619 
986651 
986683 
986714 
986746 
986778 
98t)809 
986841 
986873 


'I'ang. 


10.571948 
572453 
572959 
573466 
573973 
574481 
574989 
575497 
576007 
576516 
577026 

10.577537 
578048 
578500 
579073 
579585 
580099 
580613 
581127 
581642 
582158 

10.582674 
583190 
583707 
584225 
584743 
585262 
585781 


I'Pl" 


586821 
587342 

10.5878()3 
588385 
588908 
5^9431 
589955 
590479 
591004 
591529 
592055 
592581 

10.593108 
593636 
694164 
694692 
595222 
595751 
596282 
596813 
597344 
597876 

10.598409 
598942 
599476 
600010 
600515 
601081 
601617 
6021.34 
602691 
603229 


8.43 

8.43 

.44 

8.45 
8.46 
8.47 
8.48 
8.48 
8.49 
8.50 
8.51 
8.52 
8.53 
8.54 
8.55 
8.55 
8.56 
8.57 
8.58 
8.59 
8.60 
8.61 
8.62 
8.63 
8.64 
8.64 
8.65 
8.66 
8.67 
8.68 
8.69 
8.70 
8.71 
8.72 
8.73 
8.74 
8.74 
8.75 
8.76 
8.77 
8.78 
8.79 
8.80 
8.81 
8.82 
8.83 
8.84 
8.85 
8.86 
8.87 
8.88 
8.89 
8.90 
8.91 
8.92 
8.93 
8.94 
8.95 
8.96 
8.96 


60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 

a3 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

18 

12 

11 

10 

9 

8 

7 

ti 

5 

4 

3 

2 

1 

0 


PPI' 


409 


14< 


•760 


TABLE   IV.— LOGARITHMIC 


.y^c 


9.98G904 
986936 
986967 
986998 
987030 
987061 
987092 
987124 
9S7155 
987186 
987217 

9.987248 
987279 
987310 
987341 
987372 
987403 
987434 
987465 
987496 
987526 

9.987557 


987618 
987649 
987679 
987710 
987740 
987771 
987801 
987832 
9.987862 
987892 
987922 
987953 
987983 


988073 
988103 
988133 
.988163 
988193 


988252 


988312 
988342 
988371 
988401 


.988460 
988489 
988519 


988724 


i'an^.   PPl"  M 


10.003229 
603707 
G04306 
C04846 
G05386 
C05027 
G0G409 
607011 
G07553 
C0S097 
C08640 

10.G091S5 
609730 
610276 
610822 
C11309 
611916 
612464 
613013 
613562 
614112 

10.614663 
615214 
615766 
616318 
616871 
617425 
617980 
618;534 
619090 
619646 

10.620203 
620761 
621319 
621878 
622437 


623558 
624119 
624681 
625244 
10.62.5807 
626371 
626936 
627501 
628067 
6286a3 
629201 
629768 
630337 


10.631476 
632047 
632618 
633190 
633763 
634336 
634910 
635485 
636060 


rPl"  CotanK.  PPl 


8.97 
8.98 
8.99 
9.00 
9.01 
9.02 
9.03 
9.04 
9.05 
9.06 
9.07 
9.08 
9.09 
9.10 
9.11 
9.12 
9.13 
9.14 
9.15 
9.17 
9.18 
9.19 
9.20 
9.21 
9.22 
9.23 
9.24 
9.25 
9.26 
9.27 
9.28 
9.29 
9.30 
9.31 
9.32 
9.33 
9.34 
9.35 
9.37 


.40 

9.41 
9.42 
9.43 
9.44 
9.45 
9.46 
9.48 
9.49 
9.50 
9.51 
9.52 
9.53 
9.54 
9.55 
9.57 
9.58 
9.59 
9.60 


51 


M. 


Sine. 


.988724 
988753 
9S8782 
988811 
988840 


9SSS98 
9SSC27 
9SS950 
9S8985 
9S9014 

9.989042 
989071 
9S9100 
9S9128 
989157 
9891SG 
989214 
989243 
989271 
989300 

9.989328 
989356 
989385 
989413 
989441 
989409 
989497 
989525 
989553 
989582 

9.989610 
989637 


PPl" 


989721 
989749 
989777 
989804 1 
989832 i 
989860! 
9.9898871 
989915 
989942 
989970 


.49 
.49 

.48 
.48 
.48 
.48 
.48 
.48 
.48 
.48 
.48 
.48 
.48 
.48 
.48 
.48 
.48 
.47 
.47 
.47 
.47 
.47 
.47 
.47 
.47 
.47 
.47 
.47 
.47 
I  .47 
I  .47 
.47 
.47 
.46 
.46 
.46 
!  .46 
.46 


990025 
990052 
990079 
99010 
990134 
).  990161 
990188 
990215 
990243 
990270 
990297 
990324 
990351 


.46 
.46 
.46 
.46 
.46 
.46 
.46 
.46 
.45 
.45 
.45 
.45 
.45 
.45 
.45 
.45 
.45 
.45 
.45 


Tariff.   PPl"  M 


10.636636 
637213 
637790 
6388G8 
638947 
639520 
640107 
640687 
6412G9 
641851 
642434 

10.643018 
643602 
644187 
644773 
645360 
645947 
646535 
647124 
647713 
648303 

10.648894 
649486 
650078 
650671 
6512&5 
651859 
652455 
653051 
653647 
654245 

10.654843 
655442 
656042 
656642 
6572431 
6578451 
6584481 
659052| 
6596561 
660261 

10.660867 
661473 
662081 
6626891 
6632981 
663907! 
664518 
665129 
665741 
666354 

10.666967 
667582 
668197 
668813 
669430 
670047 
670666 
671285 
671905 
672525 


9.61 

9.62 
9.63 
9.65 
9.6G 
9.C7 
9.68 
9.  CD 
9.70 
9.71 
9.73 
9.74 
9.75 
9.70 
9.77 
9.79 
9.80 
9.81 
9.82 
9.83 
9.85 


9.87 
9.88 
9.90 
9.91 
9.92 
9.93 


9.96 

9.97 

9.98 

9.99 
10.00 
10.02 
10.03 
10.04 
10.06 
10.07 
10.081 
10. 10 1 
lO.llj 
10.121 
10.13  - 
10.15  ^ 


59 
58 
57 
56 
55 
51 
I  53 

ro 

51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 


PPl"    CotariK.    PPl"    ]M 


10. 1() 
10.17 
10.19 
10.20 
10.21 
10.23 
10.24 
10.25 
10.26 
10.28 
10.29 
10.30 
10.32 
10.33 
10.35 


13' 


410 


12* 


TS" 


SINES  AND  TANGENTS. 


19' 


.990404 
990431 
990458 
990483 
990511 
990538 
990565 
990591 
990(318 
990645 
990671 


990724 
990750 
990777 


990829 
990a5.5 
990882 
990908 
990934 
.990960 


991012 
991038 
991064 
991090 
991115 
991141 
991167 
991193 

.991218 
991244 
991270 
991295 
991321 
991346 
991372 
991397 
991422 
991448 

.991473 
991498 
991524 
991549 
991574 
901599 
991624 
991649 
991674 


.991724 
991749 
901 
991799 
991823 
901848 
901873 
99189 
991922 
991947 


PPl"   Tan!?.   PPl"  M. 


10.672525 
673147 
673769 
674393 
675017 
675642 
676267 
676894 
677521 
678149 
678778 

10.679408 


681303 


682570 
683205 
683841 
684477 
685115 
10.685753 


.n.   Co.=i 


687032 
687673 
688315 
688958 
689601 
690246 
690891 
691537 
10.692184 
692832 
693481 
694131 
694782 
69.5433 
696086 
696739 


10.698705 
699362 
700020 
700678 
701338 
701999 
702601 
703323 
703987 
704651 

10.705316 
705983 
706650 
707318 
707987 
708058 
709329 
710001 
710674 
711348 


10.36 

10.37 

10.39 

10.40 

10.41 

10.43 

10.44 

10.45 

10.47 

10.48 

10.50 

10.51 

10.53 

10.54 

10.55 

10.57 

10.58 

10.60 

10.61 

10.62 

10.64 

10.6.5 

10.67 

10.68 

10.70 

10.71 

10.73 

10.74 

10.75 

10.77 

10.78 

10.80 

10.81 

10.83 

10.84 

10.86 

10.87 

10.89 

10.90 

10.92 

10.93 

10.95 

10. 

10.98 

11.00 

11.01 

11.03 

11.04 

11.06 

11.07 

11.09 

11.11 

11.12 

11.14 

11.15 

11.1 

11.18 

11.20 

11.22 

11.23 


SliiP.     I  PPl' 


.991947 
991971 
991996 
992020 
992044 


992093 
992118 
992142 
992166 
992190 
.992214 
992239 
992263 
992287 
992311 
992335 


992406 
992430 
.992454 
992478 
992501 
992525 
992549 
992572 


992619 
992643 
992666 
.992690 
992713 
992736 
992759 
992783 
992806 
992829 
992852 
992875 


.992921 
992944 
99296' 
992990 
993013 
993036 
993059 
993081 
993104 
993127 

.993149 
9931 
993195 
993217 
993240 
993262 
993284 


2 
1 
0 
PPl"    rotans.    PPl"    IM.      M.      Co!»ine.     PPl"    Cotang.    PPl"    M. 


993329 
993351 


Tan-. 


10.711348 
712023 
712699 
713376 
714053 
714732 
715412 
716093 
716775 
717458 
718142 

10.718826 
719512 
720199 
720887 
721576 
722266 
722957 
723649 
724342 
725036 

10.725731 
726427 
727124 
727822 
728521 
729221 
729923 
730625 
731329 
732033 

10.732739 
733445 
73415:3 
734862 
735572 
736283 
736995 
737708 
738422 
739137 

10.739854 
740571 
741290 
742010 
742731 
743453 
744176 
744900 
745626 
746352 

10.747080 
747809 
748539 
749270 
750002 
750736 
751470 
752206 
752943 
753681 


PPl" I  m. 


60 


11.25 

11.26 

11.28 

11.30 

11.31 

11.33 

11.35 

11.36 

11.38 

11.40 

11.41 

11.43 

11.45 

11.47 

11.48 

11.50 

11.51 

11.53 

11.55 

11.57 

11.58 

11.60 

11.62 

11.64 

11.65 

11.67 

11.69 

11.70 

11.72 

11.74 

11.76 

11.78 

11.79 

11.81 

11.83 

11.85 

11.87 

11. 

11.90 

11.92 

11.94 

11.96 

11.98 

12.00 

12.01 

13.03 

12.05 

12.07 

12.09 

12.11 

12.13 

12.15 

12.17 

12.18 

12.20 

12.22 

12.24 

12.26 

12.28 

12.30 


11« 


4ii 


lO' 


so< 


TABLE   IV.— LOGARITHMIC 


81< 


SiJie. 


0 
1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
2.5 
20 
27 
28 
29 
30 
31 
32 
33 
34 
3.5 
30 
37 
38 
39 
40 
41 
42 
43 
44 
43 
40 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.993:351 
993;374 
993396 

993  ns 

993440 
993462 
993484 
993506 

,  993528 
993550 
993572 

9.99;i594 
993016 
993638 
9930G0 
993081 
993703 
993725 
993746 
993768 
993789 

9.993811 
993832 


993875 
993897 
993918 


994003 
.994024 
994045 
994066 
994087 
994108 
994129 
994150 
994171 
994191 
994212 
.994233 
994254 
994274 
994295 
994316 
994336 
994357 
994377 
994398 
994418 
.994438 
994459 
994479 


994519 
994-540 
994560 
991580 
994600 
994G20 


.36 


Tai 


PPl 


10.753681 
7.54421 
7.55161 
755903 
756646 
757390 
758135 
758882 
759629 
760378 
701128 

10.701880 
702632 


76414] 
761897 
765655 
766414 
707174 
707935 


10.769461 
770227 
770993 
771761 
772529 
773300 
774071 
774844 
775618 
770393 

10.777170 
777948 
778728 
779508 
780290 
781074 
781858 
782044 
783432 
78^220 

10.785011 
785802 
780595 
787389 
788185 


789780 
790580 
791381 
792183 
10.792987 
793793 
791600 
79.5408 
790218 
79T029 
707841 
7986.5,5 
799171 
800287 


; 12.32 
1 12.34 
12.36 
12.38 
12.40 
12.42 
12.44 
12.46 
12.48 
12.50 
12.52 
12.54 
12.56 
12.58 
12.60 
12.62 
12.65 
12.67 
12.69 
12.71 
12.73 
12.75 
12.77 
12.79 
12.81 
12.84 
12.86 
12.88 
12.90 
12.92 
12.94 
12.97 
12.99 
13.01 
13.03 
13.00 
13.08 
13.10 
13.12 
13.15 
13.17 
13.19 
13.21 
13.24 
13.26 
13.28 
13.31 
13.33 
13.35 
13.38 
13.40 
13.42 
13.45 
13.47 
13.49 
13.52 
13.54 
13.57 
13.59 
13.61 


PPl'  ('otane. 


5 
4 
3 
2 
1 
0 

>pi"  m7 


Sii 


9.994G20 
994040 
994000 
994080 
994700 
994720 
994739 
994759 
994779 
994798 
994818 

9.994&38 
994857 
994877 
994890 
994910 
9949X5 
994955 
994974 
994993 
995013 

9.995032 
995051 
993070 
9950S9 
995108 
993127 
995146 
995165 
993184 
995203 

9.995222 
995241 
995260 
995278 
995297 
995316 
995334 
995X53 
995372 
995390 

9.99.5109 
995427 
995446 
995464 
995482 
995501 
99.5519 
995537 
9955.55 
99.5573 

9.995591 
993610 
993028 
995046 
99.5004 
995081 


99.5717 
99.57.35 
99.57.5;^ 


PP 


..33 
.m 
.,33 
.33 
.3,3 
.33 
.33 
..33 
.33 
.33 
.33 
.33 
.33 
..32 
.,32 
.32 
.32 
.32 
.32 
.32 
.32 
.32 
.32 
.32 
.32 
.32 
.32 
.32 
.32 
.32 
.32 
.31 
.31 
.31 
.31 
.31 
.31 
.31 
.31 
.31 
.31 
.31 
.31 
.31 
.,31 
.31 
.30 
.30 
.30 
.30 
.30 
.,30 
.30 
.30 
.30 
.30 
.30 
.30 
.30 


10.800287 
801100 
801926 
802747 
803570 
804394 
80,5220 
806047 
800876 
807706 
808538 

10.809,371 
810206 
811042 
811880 
812720 
813561 
814403 
815248 
810093 
816941 

10.817789 
818640 
819492 
820345 
821201 
822058 
822916 
823776 
824638 
825501 

10.826366 
827233 
828101 
828971 
829843 
830716 
831591 
832468 
833346 
834226 

10.835108 
835992 
836877 
837764 


839543 
840435 
841329 
842225 
843123 
10.844022 
844923 
845826 
846731 
847637 
848546 
849456 
850368 
851282 
852197 


PPl'  I  Ootang.  PPl"  M 


13.64 

13. 

13. 

13.71 

13.74 

13.76 

13.79 

13.81 

13.84 

13. 

13.89 

13.91 

13.93 

13.90 

13.99 

14.02 

14.04 

14.07 

14. 

14.12 

14.15 

14.17 

14.20 

14. 

14.25 

14.28 

14.31 

14.33 

14.36 

14. 

14.42 

14.44 

14.47 

14.50 

14.53 

14.55 

14.58 

14.61 

14.64 

14.67 

14.70 

14.73 

14.76 

14.79 

14.81 

14.84 

14.87 

14.90 

14.93 

14.96 

14.99 

15.02 

15.05 

15.08 

15.11 

15.14 

15.17 

15.20 

15.23 

15.26 


9« 


412 


82< 


SINES  AND  TANGENTS. 


83° 


M. 
0 

1 
2 
3 
4 
5 


Sine.       PPl"      Tang.      PPl"     M 


.995753 
995771 

995788 


995823 
995841 


995876 
995894 


995928 
.995946 
995963 


99S015 
996032 
996049 
9960J6 
9930S3 


9.996117 
996131 
996151 
996168 
996185 
998202 
996219 
996235 
996252 
996269 

9.99628.5 
996302 
996318 
996335 
996351 
996368 
996384 
996400 
996417 
996433 

9.996449 
996465 
996482 
996498 
996514 
996530 
996546 
996562 
996578 
996594 

9.996010 
996625 
996641 
996657 
990673 


996704 
996720 
9967;55 
996751 


.29 
.29 
.29 
.29 
.29 
.29 
•29 
.29 
.29 
.29 
.29 
.29 
.29 
.29 
.29 
.29 
.29 
.29 
.29 
.28 
.2S 
.2S 
.28 
.28 
.23 
.28 
.28 
.28 
.28 
.28 
.28 
.28 
.27 
.27 
.27 
.27 
.27 
.27 
.27 
.27 
.27 
.27 
.27 
.27 
.27 
.27 
.27 
.27 
.27 
.26 
.26 
.26 
.26 
.26 
.26 
.26 
.26 
.26 
.23 


10.8.52197 
&53115 
a54034 
854956 
855879 


857731 
858660 
859591 
860524 
8614.58 

10.862395 
86333:3 
864274 
865216 
866161 
867107 
868056 
889006 
889959 
870913 

10.871870 
872828 
873789 
874751 
875716 
876683 
877652 
878623 
879596 
880.571 

10.881548 
882528 
883509 
884493 
885479 
88()467 
887457 
888149 
889444 
8J0141 

10.891140 
892441 
893444 
894150 
895158 
896468 
897481 
898496 
899513 
9005.32 

10.901.554 
902578 
903605 
901633 
905664 


PPl" 
35*" 


907734 

908772 


8108.56 


15.29 
15.32 
15.35 
15.39 
15.42 
15.45 
15.48 
15.51 
15.55 
15.58 
1.5.61 
15.64 
15.67 
1.5.71 
15.74 
15.77 
1.5.81 
1.5.84 
15.87 
91 
15.94 
5.97 
16.01 
16.04 
18.07 
10.11 
1{;.15 
16.18 
16.22 
16.25 
16.29 
16.32 
16.36 
16.39 
16.43 
16.46 
16.50 
16.54 
16.58 
16.61 
16.65 
16. 
16.72 
16.76 
16.80 
16.84 
16.87 
16.91 
16.95 
16.99 
17.03 
17.07 
17.11 
17.15 
17.19 
17.22 
17.27 
17.30 
17.34 
17.38 


PPl' 


9.996751 
996766 
996782 
996797 
996812 
996828 
996^43 
996858 
996874 


996934 
996^9 


996979 


997009 
997024 


997053 

9.997068 

997083 


997112 
997127 
997141 
997156 
997170 
997185 
997199 

9.997214 
997228 
997242 
997257 
997271 
997285 
997299 
997313 
997;^ 
997341 

9.997355 
997369 
997383 
997397 
997411 
997425 
997439 
997452 


997480 
.997493 
997507 
997520 
997534 
997547 
997561 
997574 
997588 
997601 
997614 


.26 
.26 
.26 
.26 
.2,5 
.25 
.25 
.25 
.2.5 
.25 
.25 
.25 
.25 
.25 
.25 
.25 
.2.5 
.25 
.25 
.25 
.25 
.24 
.24 
.24 
.24 
.24 
.24 
.24 
.21 
.24 
.24 
.24 
.24 
.24 
.24 
.24 
.24 
.24 
.23 
.23 
.23 
.23 
.23 
.2:3 
.2:3 
.23 
.23 
.23 
.23 
.23 
.2:3 
.23 
.23 
.22 
.22 
.22 
.22 
.22 
.22 


'Iju 


10.910856 
911902 
912950 
914000 
915053 
916109 
917167 
918227 
919290 
920356 
921424 

10.922495 
92a568 
924644 
925722 
926803 
927887 
928973 
9,30062 
9311.54 
932248 

10.933345 
934444 
935547 
936652 
937760 
938870 


PPl" 


941100 
942219 
943341 
10-944465 
945593 
&46723 
ft47856 


Cosine.  |PPr 


950131 
951273 
952418 
953566 
954716 

10.9.55870 
957027 
958187 
959349 
9<,a515 
961G84 
962856 
964031 
965209 
966391 

10.967575 
968763 


971148 
972345 
973545 
974749 
975956 
977166 
978380 


17.43 

17.47 

17.51 

17.55 

17.59 

17.63 

17.67 

17.72 

17.76 

17.80 

17.84 

17.89 

17.93 

17.97 

18.02 

18.06 

18.10 

18.15 

18.19 

18.24 

18.28 

18.33 

18.37 

18.42 

18.46 

18.51 

18.55 

18.60 

18.65 

18.70 

18.74 

18.79 

18.84 

18.89 

18.93 

18.98 

19.03 

19.08 

19.13 

19.18 

19.23 

19.28 

19.3:3 

19.38 

19.43 

19.48 

19.53 

19.58 

19.64 

19. 

19.74 

19.79 

19.85 

19.90 

19.95 

20.00 

20.06 

20.11 

20.17 

20.23 


Cotang. 


PPl"  I  M. 


60 


84" 


TABLE   IV.— LOGARITHMLC 


85° 


0 

1 

2 
3 
4 


7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.997614 
997628 
997641 
997^54 
997667 


997706 
997719 

997732 
997745 

.997758 
997771 
997784 
997797 
997809 
997822 
9978;^ 
997847 
997860 
997872 

.997885 


997910 
997922 
99793.5 
997947 
997959 
997972 
997984 


998020 
998032 


998056 


998080 
998092 
998104 
998116 

.998128 
9981  :J9 
998151 
998163 
998174 
998186 
998197 
998209 
998220 
998232 

.998243 
998255 
998266 
998277 


998300 
998311 
998322 


998344 


Tan' 


.978380 
979597 
980817 
982041 
98:^268 
984498 
985732 


10. 


988210 
989454 
990702 
991953 
993208 
994466 
995728 
996993 
998262 
999535 
000812 
002092 
003376 
004663 
005955 
007250 
008549 
009851 
011158 
012468 
013783 
015101 
016423 
017749 
019079 
020414 
021752 
023094 
024440 
025791 
027145 
028504 
029867 
031234 
032606 
033981 
035361 
036745 
0:38134 
039527 
040925 
042326 
043733 
045144 
046559 
047979 
049403 
0.50832 
052266 
053705 
055148 
056596 
058048 


Cosine.  PPl"  Cotaiigr.  PPl"  M. 


}'V 


20.28 
20.  S3 
20.40 
20.45 
20.51 
20.56 
20.62 
20.68 
20.74 
20.80 
20.8.5 
20.91 
20.97 
21.03 
21.09 
21.15 
21.21 
21.27 
21.. 34 
21.40 
21.46 
21.52 
21. .58 
21.65 
21.71 
21.78 
21.84 
21.91 
21.97 
22.04 
22.10 
22.17 
22.23 
22.30 
22.37 
22.44 
22.51 
22.57 
22.65 
22.71 
22.79 
22.86 
22.93 
23.00 
23.07 
23.14 
23.22 
23.29 
2:3.37 
23.44 
23.51 
23.60 
23.66 
23.74 
23.82 
2:3.90 
23.97 
24.05 
24.13 
24.21 


9.9!(8344 
99835.5 
998366 
998377 
998388 
998399 
998410 
998421 
998431 
998442 
998453 

9.998464 
998474 
998485 


998506 
998516 
998527 
998537 
998548 
998558 
).  998568 
998578 


998619 
998629 


998649 


998708 
998718 
998728 
99873 
99874' 
99875 

9.998766 
998776 
998785 
998795 
.  998804 
99b813 
99882:} 
998832 
998841 
998851 

9.998860 


998878 
998887 


998905 


998923 
998932 


rnug.  PPl"  M. 


I1.05804S 
059506 
000908 
0624:35 
OC:3907 
065384 
066866 
068a53 
069845 
071:342 
072M4 

11.074351 
075864 
077381 
078904 
0804.32 
081966 
083505 
085049 


088154 

11.089715 

091281 


094430 
096013 
097602 
099197 
10079' 
102404 
104016 

11.10.5()34 
107258 

.  108888 
110524 
112167 
113815 
115470 
1171,31 
118798 
120471 

11.1221.51 
123838 
12.55:31 
1272:30 
128936 
130(349 
132368 
134094 
135827 
137567 

11.139314 
141068 
142829 
144597 
146372 
1481.54 
149943 
151740 
153545 
155a56 


PPl"!  Cotaiig. 


24.29 
24.37 
24.45 
24.53 
24.62 
24.70 
24.78 
24.87 
24.85 
2.5.03 
2,5.12 
25.21 
25.30 
2.5.38 
25.47 
25.56 
25.&5 
25.74 
25.83 
25.92 
26.01 
26.10 
26.20 
26.29 
26.38 
26.48 
26.58 
26.67 
26.77 
26.87 
26.97 
27.07 
27.17 
27.27 
27.37 
27.47 
27.58 
27.68 
27.79 
27.89 
28.00 
28.11 
28.21 
28.32 
28.43 
28.54 
28.66 
28.77 
28.88 
29.00 
29.11 
29.23 
29.:35 
29.46 
29.58 
29.70 
29.82 
29.95 
30.07 
30.19 


PPl' 


414 


S6o_ 

3ir~ 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
;ii 
.51 
5.-) 
56 


60 


SINES  AND  TANGENTS. 


8T» 


).  998941 


998958 


998976 
998984 


999002 
999010 


999027 
.999036 
999044 
9990.5;^ 
999061 


999077 


999094 
999102 
999110 

9.999118 
999126 
999l;« 
999142 
999150 
999158 
999166 
999174 
999181 
999189 

9.999197 
999205 
999212 
999220 
999227 
99f)2;i5 
999242 
999250 
9992.)7 
99926.5 

9.999272 
999279 
999287 
999294 
999301 
999308 
999315 
999322 
999329 
999336 

9.999343 
999;j.50 
9993.57 
999364 
999371 
999378 
999384 
999391 
999398 
999404 


PFl 


'I'aii^ 


.15 
.15 
.15 
.14 
.14 
.14 
.14 
.14 
.14 
.14 
.14 
.14 
.14 
.14 
.14 
.14 
.14 
.14 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.12 
.12 
.12 
.12 
.12 
.12 
.12 
.12 
.12 
.12 
.12 
.12 
.12 
.12 
.12 
.11 
.11 
.11 
.11 
.11 
.11 
.11 


11.1.5.53.56 
1.57175 
1.59002 
160837 
162679 
164529 
166387 
1682.52 
170126 
172008 
173897 

11.17.5795 
177702 
179616 
181.539 
18;3471 
18.5411 
187359 
189317 
191283 
1932,58 

11.19.3242 
1972,3.5 
1992.37 
201248 
203269 
205299 
207338 
209;«7 
211446 
213514 

11.21.5592 
217680 
219778 
221886 
224005 
226131 
228273 
2:30422 
23258;^ 
2;i47.51 

11.2:369^5 
239128 
241:332 
2ia547 
215773 
248011 
2.)0260 
252521 
254793 
2.57078 

11.2.59:374 
20168:3 
264004 
260337 
268683 
271041 
273412 
275796 
278194 
280604 


30.32 
:30.45 

.,57 
30.70 
30.83 
30.96 
31.10 
31.23 
31.36 
31.50 
31.63 
31.77 
31.91 
:32.05 
32.19 
32.33 
32.48 
32.62 
.32.77 
32.92 
313.07 

.22 
3:3.37 
33.52 
33.68 
3:3.83 
33.99 
34.15 
;34.31 
34.47 
34.64 
34.80 
:34.97 
.3.5.14 
3.3.31 
:3.5.48 
•3;5.ft3 
:3.5.83 
36.00 
36.18 
;36.36 
36.55 
36.73 
36.92 
37.10 
37.29 
37.49 
37.68 
37.87 
38.07 
;38.27 
38.48 
38.68 
38.89 
39. 
39.30 
39.52 
39.74 
;39.95 
40.17 


PPl"  Cotaiu 


M. 


9.999404 
999411 


999424 
999431 
999437 
999443 
999450 
9994.56 
999463 
999469 
.999475 
999481 
999487 
999493 
999500 


999512 
999518 
999524 
999.529 
.99953.5 


9995.58 
999564 
999570 
95)9.575 
999581 
999,58(i 

9.999-592 
999597 
9'j9<i03 
999608 
9it96l4 
99fJ619 
999024 
999629 
99iMi;3.: 
999640 

9.999645 
999050 
99965^3 
99{Ki(.0 
95)96(^3 
999670 
9r)9()75 
99i,'680 
999085 
999689 

9.99i)094 
999699 
999704 
999708 
999713 
999717 
999722 
999726 
999731 
999735 


Cosii.*-.      PPl"     Vot 


Tans. 


11.280004 
283028 
285466 
287917 
2i)0382 
292860 
295:3.54 
297861 
300.383 
302919 
305471 

11.8080:37 
310619 
313216 
315828 
3184.36 
321100 
323761 
326437 
329i;30 
331840 

ll.;3345«)7 
337311 
ai0072 
342851 
345W8 
348463 
351296 
&54147 
357018 
a59907 

11.362816 
365744 
368692 
371060 
3",4048 
37765' 
380687 
3837138 
38681 


11.393022 
396161 
399323 
402508 
40571 
408949 
412205 
415486 
418792 
422123 

11.42.5480 
428863 
432273 
43571 
439172 
442664 
446183 
449732 
453309 
456916 


PPl' 


40.40 
40.62 
40.85 
41.08 
41.32 
41.55 
41.79 
42.0:3 
42.28 
42.. 32 
42.77 
43.03 
43.28 
43.54 
43.80 
44.07 
44.34 
44.61 
44.88 
45.16 
45.44 
45.73 
46.02 
46.31 
46.61 
46.91 
47.22 
47.53 
47.84 
48.16 
48.48 
48.80 
49.13 
49.47 
49.81 
.30.15 
50.50 
50.85 
51.21 
.51. .58 
51.94 
52.-32 
52.70 
53.08 
53.47 
53.87 
54.27 
54.68 
55.10 
55.52 
55,95 
56.38 
56.82 
57.27 
57.73 
58.19 
.58.66 
59.14 
59.62 
60.12 


PPl"   M. 


iJ° 


415 


2« 


88< 


TABLE   IV.— SINES  AND  TANGENTS. 


89° 


Siji' 


9.9997a5 
999740 
999744 
999748 
99975;^ 
999757 
999761 
999765 
999769 
999774 
999778 

9.999782 
999786 
999790 
999794 
999797 
999801 
999805 
999809 
999813 


9.999820 
999824 
999827 
999&M 
999834 
999838 
999841 
999844 
999848 
999851 

9.999854 
9998.58 
999861 
999864 
999867 
999870 
999873 
99987f) 
999879 
999882 

9.999885 
9998.S8 
9f)9891 


999897 
999899 
999902 
999905 
999907 
999910 
).  999913 
999915 
9<)9918 
999920 
99^)22 
99992.5 
999927 
999})29 
9i)f»9;}2 
999934 


(/osi 


.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.06 
.06 
.06 
.06 
.06 
.06 
.06 
.06 
.06 
.06 
.06 
.06 
.06 
.06 
.06 
.05 
.05 
.05 
.05 
.05 
.05 
.05 
.05 
.05 
.05 
.05 
.05 
.05 
.05 
.05 
.05 
.01 
.01 
.01 
.04 
.04 
.01 
.04 
.01 
.04 
.04 
.04 
.01 
.04 
.04 


Tai.g. 


11.4.56916 

46a5.5;5 

461221 
467920 
471(;51 
475414 
479210 
483039 
486902 
490800 
494733 

11.498702 
50270 
506750 
510*^0 
514950 
519108 
52.3.307 
527546 
531828 
536151 

11.. 540519 
544930 
549;« 
5.5;«90 
558440 
5630;^ 
567685 
572:382 
577131 
581932 

11.. 586787 
591691) 
596662 
601685 
606766 
611908 
617111 
622.378 
627708 
6;«105 

11.638.570 
644105 
649711 
655390 
661144 
666975 
672886 
678878 
6849;>4 
691116 

11.697,366 
703708 
710144 
716677 
72.3;»9 
7:30044 
736885 
7438a5 


758()7i 


PPI"    (V)taim.    PIM"    M 


I'l' 


60.62 
61.13 
61.65 
62.18 
62.72 
63.26 
63.82 
64.:-}9 
64.96 
65.55 
66.15 
66.76 
67.38 
68.01 
68.6.5 
69.31 
69.98 
70.6(5 
71. a5 
72.06 
72.79 
73.52 
74.28 
75.05 
75.83 
76.63 
77.45 
78.29 
79.14 
80.02 
80.91 
81.82 
82.76 
83.71 
84.70 
85.70 
86.72 
87.77 
88.  &5 
89.95 
91.08 
92.24 
93.43 
94.65 
9.5.90 
97.19 
98.51 
99.87 
101.3 
102.7 
104.2 
105.7 
107.2 
108.9 
110.5 
112.2 
114.0 
115.8 
117.7 
119.7 


9.999934 
9999:36 
999938 
999940 
999942 
999944 
999946 


999950 
999952 
9999,54 
9.9999;56 
999958 


999961 
999963 
999964 
999966 
9999()8 
9999(i9 
999971 
9.999972 
999973 
999975 


999979 


999981 
999982 


999988 


99{}992 
999993 

9.999993 
999994 
999{)95 
999995 
99999(1 
999996 
999997 
999997 
999998 
999998 

9.999999 


999999 

999999 

10.000000 

000000 

mmo 

000000 
OOWKX) 
000000 


Cosiiif 


1>P1"      Tii 


PPI 


11.7,5^079 
765:379 
77280.5 
780:359 
788047 
795874 
803844 
811964 
8;^02:37 
828672 
837273 

11.846048 
8.55004 
864149 
873490 
8a30:37 
892797 
902783 
9i;j(K)3 
923409 
934194 

11.94.5191 
9.56473 
968055 
979956 
992191 

12.004781 
017747 
031111 
044900 
059142 

12.07:3866 
089106 
101901 
121292 
138326 
l,5()a56 
174540 
19.3845 
214049 
2:3,5239 

12.257516 
280997 
1305821 
3.32151 
360180 
390143 
422328 
457091 
494880 
536273 

12.,5820:30 
633183 
091175 
758122 
837304 
9:34214 

13.0.59153 
2:3.5244 
5:36274 

Infinite. 


("otunt 


PPI" 


121.7 

123.8 

125.9 

128.1 

1:30.4 

1,32.8 

135.3 

1.37.9 

140.6 

143.3 

146.2 

149.3 

152.4 

155.7 

1,59.1 

162.7 

166.4 

170.3 

174.4 

178.7 

18:3.3 

188.0 

193.0 

198.3 

20:3.9 

209.8 

216.1 

222.7 

229.8 

237.3 

245.4 

2.54.0 

263.2 

273.2 

283.9 

295.5 

308.0 

321.7 

336.7 

3.53.2 

371.2 

;391.3 

413.7 

4:38.8 

467.1 

499.4 

.5:36.4 

.579.4 

629.8 

689.9 

762.6 

8;52.5 

966.5 

1116 

1320 

1615 

2082 

29:35 

5017 


PPI"    M 


410 


TABLE  Y. 

PRECISE  CALCULATION  OF  FUNCTIONS. 

The  proportional  parts,  as  given  in  Table  IV,  are  sufficient  for 
ordinary  use.  When  precision  is  desired  the  following  rules  should 
be  observed : 

I.  In  finding  the  logarithmic  function  of  an  angle  expressed  in 
degrees,  minutes,  and  seconds,  derive  it  from  that  function  which  is 
nearest  to  it,  whether  greater  or  less;  for,  the  proportional  parts, 
being  only  approximations,  should  be  multiplied  by  as  small  a 
number  as  possible. 

II.  In  finding  the  angle  from  its  given  function,  use  that  loga- 
rithm which  differs  least  from  the  one  given,  subtracting  or  adding 
as  the  case  may  be. 

III.  To  find  the  logarithmic  sine  of  an  angle  of  less  than  2°  36^ : 
reduce  it  to  seconds;  add  the  logarithm  of  the  number  of  seconds 
to  the  logarithmic  sine  of  one  second,  which  is  4.685575;  from 
this  sum  subtract  the  difierence  in  the  following  table  correspond- 
ing to  the  number  of  seconds ;  the  remainder  is  the  required  loga- 
rithmic sine  within  one  millionth.  * 

IV.  Conversely,  to  find  the  angle  when  the  given  logarithmic 
sine  is  less  than  8.656702:  first,  find  tlie  angle  approximately  by 
Table  IV;  reduce  this  to  seconds;  add  to  the  given  sine  the  differ- 
ence in  the  following  table  corresponding  to  the  number  of  sec- 
onds; from  this  sum  subtract  4.685575;  the  remainder  is  the  loga- 
rithm of  the  required  number  of  seconds  within  one. 

V.  To  find  the  logarithmic  tangent  of  an  angle  less  than  2°  36'' : 
reduce  it  to  seconds;  add  to  the  logarithm  of  the  number  of  sec- 
onds the  logarithmic  tangent  of  one  second,  which  is  4.685575;  to 
this  sum  add  the  difference  in  the  table  (p.  419  and  420)  corres- 
ponding to  the  number  of  seconds ;  the  sum  is  the  required  loga- 
rithmic tangent  within  one  millionth. 

VI.  To  find  the  angle  when  the  given  logarithmic  tangent  is  less 
than  8.657149,  which  is  the  tangent  of  2°  36':  first  find  the  angle 
approximately  by  Table  IV;  reduce  it  to  seconds;  subtract  from 
the  given  tangent  the  difference  in  the  table  corresponding  to  the 
number  of  seconds;  from  this  remainder  subtract  4.685575;  the 
remainder  is  the  logarithm  of  the  required  number  of  seconds 
within  one. 

VII.  To  find  the  logarithmic  cotangent  of  an  angle  less  than  2° 
36' :  reduce  it  to  seconds ;  subtract  the  logarithm  of  the  number  of 
seconds  from  the  logarithmic  cotangent  of  one  second,  which  is 
15.314425;  from  this  remainder  ^btract  the  difference  in  the 
table  corresponding  to  the  number  of  seconds;  the  remainder  is 
the  required  logarithmic  cotangent  within  one  millionth. 

VIII.  To  find  the  angle  when  the  given  logarithmic  cotangent  is 
greater  than  11.342851,  the  cotangent  of  2°  36':  first  find  the  angle 
approximately  by  Table  IV ;  reduce  it  to  seconds ;  add  to  the  given 
cotangent  the  difference  in  the  table  corresponding  to  the  number 
of  seconds;  subtract  this  sum  from  15.314425;  the  remainder  is 
the  logarithm  of  the  required  number  of  seconds  within  one. 


417 


TABLE  v.— AIDS  TO 


FOR    THE   SINES    OF    SMALL    ANGLES. 


Angles.       Seconds.    Diff, 


0" 
9' 
15'  50" 

20'  20" 
23'  50" 

27' 

29'  50" 
32'  30" 
35' 

37'  20" 
39'  30" 

41' 30" 
43'  20" 
45'  10" 
47' 
48'  40" 

50'  20" 
52' 

53'  30" 
55' 
56'  30" 

58' 

59'  20" 
1°  00'  40" 

2' 

3'  20" 


4'  40" 
5' 50" 
7' 

8'  10" 
9'  20" 

10'  30" 
11'  40" 
12'  50" 
14' 
15' 

16'  10" 
17'  10" 
18'  10' 
19'  20" 
20'  20" 

21'  20" 
22  20" 
23'  20" 
24'  20" 
2.3'  10" 

26'  10" 
27'  10" 
28'  10" 
29' 
29' 60" 


0 
540 

950 
1220 
H30 
1620 

1790 
1950 
2100 
2240 
2370 

2490 
2600 
2710 
2820 
2920 

3020 
3120 
3210 
3300 
3390 

3480 
3-^60 
3640 
3720 
3800 


39.50 
4020 
4090 
4160 

4230 
4300 
4370 
4440 
4.300 

4570 
4630 
4690 
4760 
4820 

4880 
4940 
5000 
50(10 
5110 

5170 
52.-50 
5290 
6340 


AngU'S. 

Second s. 

Dili. 

10  29'  50" 

5390 

50 
51 

52 
53 
54 

SO'  50" 

5450 

31'  40" 

5500 

32'  ,30" 

5550 

33'  30" 

5C10 

34'  20" 

5G60 

oo 

35'  10" 

5710 

56 

57 
58 
59 
CO 

36' 

5760 

36'  50" 

5810 

37'  40" 

58(i0 

38'  30" 

5910 

39'  30" 

5970 

61 
62 
63 
64 
65 

40'  20" 

60i:o 

41'  10" 

6070 

41'  50" 

6110 

42'  40" 

6160 

43'  30" 

6210 

66 

67 
68 
69 
70 

44'  10" 

6250 

45' 

6300 

45'  50" 

63.50 

46'  30" 

6390 

47'  20" 

6440 

71 
72 
73 
74 
75 

48' 

6480 

48'  ,50" 

6530 

49'  30" 

6570 

50'  20" 

6620 

51' 

6660 

76 

77 
78 
79 
80 

51'  50" 

6710 

52'  30" 

6750 

63'  10" 

6790 

54' 

6840 

54'  40" 

6880 

81 
82 
83 
84 
85 

55'  20" 

6920 

56'  10" 

6970 

56'  50" 

7010 

57' 30" 

7050 

58'  10" 

7090 

86 
87 
88 
89 
90 

58'  50" 

7130 

59'  ,30" 

7170 

2°  00'  10" 

7210 

50" 

7250 

1'  40" 

7.300 

91 
92 
93 
94 
95 

2' 20" 

7310 

3'  " 

7380 

3'  S5" 

7415 

4'  10" 

7450 

4'  50" 

7490 

96 
97 
98 
99 

6'  30" 

75^30 

6'  10" 

7570 

6'  50" 

7610 

7' 30" 

7650 

Angles.       Seconds.     Diff, 


2° 


7'  30" 
8'  10" 
8'  45" 
9' 20" 

10' 

10'  40" 

11'  15" 
11'  50" 
12'  30" 
13'  5" 
13'  40" 

14'  20" 
15' 

15'  35" 
16'  10" 
16'  46" 

17'  20" 
17'  56" 

18'  30" 
19'  5" 
19'  40" 

20'  15" 
20'  50" 
21'  25" 
22' 
22' 35" 

23'  10" 
23'  45" 
24'  20" 
24'  55" 
25'  30" 

26' 

26'  35" 
27'  5" 

27'  40" 
28'  10" 

28'  45" 
29'  15" 
29'  50" 
30'  20" 
SO'  55" 

31'  25" 
32' 

32'  30" 
33'  5" 
33'  35" 

34'  5" 
34'  40" 
35'  10" 
35'  40" 
36'  15" 


7650 
7690 
7725 
7760 
7800 
7840 

7875 
7910 
7950 
7985 
8020 

8060 
8100 
8135 
8170 
8206 

8240 
8275 
8310 
8345 


8415 
8450 
8486 
8520 
8666 

8590 
8625 
8660 
8695 
8730 

8760 
8795 

8825 


8925 
8955 
8990 
9020 
9055 


9120 
9150 
91  &5 
9216 

9245 

9280 
9310 
9340 
9375 


418 


PRECISE   CALCULATIONS. 


FOB  TANGENTS  AND  COTANGENTS  OF  SMALL,   ANGLES. 


Angk 


0" 

7'  10" 
11'  10" 
14'  10" 
17' 
19' 

21' 
23' 

24' SC 
26'  30' 
27' 50' 

29' 2^ 
30'  40' 
32' 

33'  10' 
34'  20' 

35' 30' 
36'  40' 
37' 50' 
38' 50' 
39' 5C 

40' 50' 
41' 50' 
42' 50' 
43' 50' 
44' 40' 

45' 40' 
46'  30' 
47'  20' 
48' 10' 
49' 

49' 50' 
50' 40' 
51'  30' 
52'  20' 
53' 

53' 50' 
54'  40' 
55'  20' 
56' 
56' 50' 

57'  30' 
58'  10' 
58'  50' 
59'  30' 
0'20' 

1' 
1'40' 

2'  10' 
2' 50' 
3'  30' 


Seconds. 

Diff. 

0 

1 
2 
3 
4 
5 

0 
430 

670 

850 

1020 

1140 

1260 

6 

7 
8 
9 
10 

1380 

1490 

1590 

1670 

1760 

11 
12 
13 
14 
15 

1840 

1920 

1990 

2060 

2130 
2200 

16 
17 
18 
19 
20 

2270 

2330 
2390 

2450 

21 
22 
23 
24 
25 

2510 
2570 
2630 
2680 

2740 
2790 

26 

27 
28 
29 
30 

2840 
2890 
2940 

2990 

31 
32 
33 
34 

35 

3040 
3090 
3140 
3180 

3230 
3280 

36 
37 
38 
39 
40 

3320 
a360 
3410 

3450 

41 
42 
43 
44 

45 

3490 
3530 
3.570 

3620 

3660 

46 
47 
48 
49 

3700 
3730 
3770 

3810 

Angles.       Seconds.    Diff. 


1° 


3' 30" 
4'  10" 
4' 50" 
5' 30" 
6' 
6'  40" 

7'  20" 
7' 50" 
8' 30" 
9' 
9'  40" 

10'  20" 
10'  50" 
11' 30" 
12' 
12' 30" 

13'  10" 
13'  40" 
14'  10" 
14'  50" 
15'  20" 

15'  50" 
16'  20" 
17' 
17' 30" 

18' 

18'  30" 
19' 

19'  30" 
20' 
20'  30" 

21' 

21' 30" 
22' 

22'  30" 
23' 

23'  30" 
24' 

24'  30" 
25' 
25'  30" 

26' 

26'  30" 
26'  50" 
27'  20" 
27' 50" 

28'  20" 
28'  40" 
29'  10" 
29'  40" 
30'  10" 


3810 
3850 
3890 
3930 
3960 
4000 

4040 
4070 
4110 
4140 

4180 

4220 
4250 
4290 
4320 
4350 

4390 
4420 
4450 
4490 
4520 

4550 
4580 
4620 
4650 
4680 

4710 
4740 
4770 
4800 
4830 


4950 


5010 
5040 
5070 
5100 
5130 

5160 
5190 
5210 
5240 
5270 

5300 
5320 
5350 
5380 
5410 


67 


87 


Angles. 


1°  30'  10' 
30' 30' 
31' 

31' 30' 
32' 
32' 20' 

32' 50' 
33' 10' 
33'  40' 
34'  10' 
34' 30' 


35' 20' 
35' 50' 
36'  10' 
36'  40' 

37'  10' 
37'  30' 
38' 

38' 20' 
38' 50* 

39'  10' 
89^30' 
40' 

40'  20' 
40' 5^ 

41'  IC 
41'  40' 

42' 

42'  30' 
42' 50' 

43'  10' 
43'  40' 
44' 

44' 30' 
44'  50' 

45'  20" 
45'  40' 
46' 

46'  20' 
40' 40' 

47'  10' 
47' 30' 
48' 

48'  20' 
48'  40' 

49' 

49'  20' 
49'  40' 
50'  10' 
50' 30' 


Seconds.  I   Diff. 


&410 
5430 
5460 
5490 
5520 
5540 

5570 
5590 
5620 
56.50 
5670 

5700 
5720 
5750 
5770 
5800 

5830 
5850 
5880 
5900 


5950 
5970 


6050 

6070 
6100 
6120 
6150 
6170 

6190 
6220 
6240 
6270 
6290 

6320 
6340 
6360 
6380 
6400 

6430 
6450 
6480 
6500 
6520 

6540 
6560 
6580 
6610 


100 
101 
102 
103 
104 
105 

106 
107 
108 
109 
110 

111 
112 
113 
114 
115 

116 
117 
118 
119 
120 

121 
122 
12:1 
124 
125 

126 
127 
128 
129 
130 

131 
132 
133 
134 
135 

136 
137 
138 
139 
140 

141 
142 
143 
144 
145 

146 
147 

148 
149 


419 


TABLE  v.— AIDS   TO    PRECISE    CALCULATIONS. 

FOR  TANGENTS  AND  COTANGENTS  OF  SMALL  ANGLES. 


Angle 


1°  50'  30' 
50' 50' 
51'  10' 
51'  30' 
52' 
52'  20' 

52'  40' 
53' 

53'  20' 
53'  50' 
54'  10' 

54'  30" 
54'  60'' 
55'  10' 
55'  30'- 
65' 50' 

56'  10' 
5G'  30' 
56'  50' 
57'  10' 
57'  40' 

58' 

58'  20' 
68'  40' 
59' 
59'  20' 

59'  40' 

2°  00'  00' 

20' 

40' 

1' 

1'20' 
1'40' 
2' 

2' 20' 
2' 40' 

3' 

3' 20' 
3' 40' 
4' 
4'  20' 

4' 40' 
6' 

5' 20' 
5' 40' 
b' 

6' 20' 
6'  40' 
7' 

7'20' 
7' 40' 


Seconds.  Diff. 


6630 
6650 
6670 
6690 
6720 
6740 

6760 
6780 
6800 
6830 
6850 

6870 
6890 
6910 
6930 
6950 


7010 
7030 
7060 

7080 
7100 
7120 
7140 
7160 

7180 
7200 
7220 
7240 
7260 

7280 
7300 
7320 
7340 
7360 

7380 
7400 
7420 
7440 
7460 

7480 
7500 
7520 
7540 
7560 

7580 
7600 
7620 
7640 
7660 


I  ir. 


156 
157 
158 
159 
160 

161 
162 
163 
164 
165 

166 
167 
168 
169 
170 

171 
172 
173 
174 
175 

176 
177 
178 
179 
180 

181 
182 
183 
184 
185 

186 
187 
188 
189 
190 

191 
192 
193 
194 
195 

196 
197 
198 
199 


Angles. 


2o   7'  40" 
8' 

8'  15' 
8' 30' 
8' 50' 
9'  10' 

9'  80' 

9' 50' 
10'  10' 
10'  20' 
10'  40' 

10'  55' 
11'  15' 
11' 35' 
11' 55' 
12'  15' 

12'  35' 
12'  55' 
13'  15' 
13'  35' 
13'  50' 

14'  10' 
14' 30' 
14'  45 
15'  5' 
15'  20' 

15'  40' 
15'  56' 
16'  15' 
16'  30' 
16'  50' 

17'    5' 
17' 25' 
17'  40' 
18' 
18'  15' 

18' 35' 
18' 55' 
19'  15' 
19'  30' 
19'  45' 

20'  5' 
20'  20' 
20'  40' 
20' 55' 
21'  15' 

21'  30' 
21'  45' 
22'  5' 
22' 20' 
22' 35' 


Seconds.    Diff 


7660 
7680 
7695 
7710 
7730 
7750 

7770 

7790 
7810 
7820 
7840 

7855 
7875 
7895 
7915 
7935 

7955 
7975 
7995 
8015 


8050 
8070 

8085 
8105 
8120 

8140 
8155 
8175 
8190 
8210 

8225 
8245 
8260 
8280 
8295 

8315 
aS35 
8355 
8370 
8385 

8405 
8420 
8440 
8455 
8475 

8490 
8j05 
8525 
8610 
8555 


200 
201 
202 
203 
204 
205 

206 
207 
208 
209 
210 

211 
212 
213 
214 
215 

216 
217 
218 
219 
220 

221 
222 
223 
224 
225 

226 
227 
228 
229 
230 

231 
232 
233 
234 

235 


237 
238 
239 
240 

241 
242 
243 
244 
245 

246 
247 
248 
249 


Anglef 


2°  22'  35" 
22' 55" 
23'  10" 
23'  30" 
23'  45" 
24' 

24'  20" 
24'  35" 
24' 55" 
25'  10" 
25'  25" 

25'  45" 
26' 

26'  20" 
26'  35" 
26'  50" 

27'  10" 
27' 25" 
27'  45" 
28' 
28'  15" 

28'  35" 
28'  50" 
29'  10" 
29'  25" 
29' 40" 


SO'  15" 
30'  30" 
30'  50" 
31'    5" 

31'  20" 
31'  35" 
31'  55" 
32'  10" 
32'  25" 

32'  40" 
32' 55" 
33'  15" 
33' 30" 
33'  45" 

34' 

34'  15" 
34'  30" 
34'  45" 
35' 

35' 20" 
35'  35" 
35' 50" 
36'  5" 
86' 20" 


8555 
8575 
8590 
8610 
8625 
8640 


8675 
8695 
8710 
8725 

8745 

8760 
8780 
8795 
8810 

8830 
8845 
8865 
8880 


8915 


8965 


9000 
9015 
9030 
9050 
9066 

9080 
9095 
9115 
9130 
9145 

9160 
9176 
9195 
9210 
9225 

9240 
9255 
9270 

9285 
9300 

9320 
9335 
9350 


420 


RETURN   EDUCATION-PSYCHOLOGY  LIBRARY 

TO— #^  2600  Tolman  Hall  642-4209 


LOAN  PERIOD 
■    1  MONTH 


ALL  BOOKS  MAY  BE  RECALLED  AFTER  7  DAYS 

2-  hour  books  must  be  renewed  in  person 

Return  to  desk  from  which  borrowed 


DUE  AS  STAMPED  BELOW 


UNIVERSITY  OF  CALIFORNIA,  BERKELEY 
FORM  NO.  DDIO,  5m,  3/80  BERKELEY,  CA  94720 

®s 


U^  K^K^    y  W^^k^ 


9S4225 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


